∫Calc Practice
Home›Calculus 1›The chain rule in Leibniz notation

The chain rule in Leibniz notation practice problems

dy/dx = (dy/du)(du/dx), with y given in terms of u and u in terms of x. 60 problems with worked solutions; in 60 of them every equation is proved by a computer algebra system.

20 worked problems

Try each one before opening the solution. Every line is labeled with what checked it. How we verify

Problem 2.2217

easy
Let \( \displaystyle y = u^{2} + u \) and \( \displaystyle u = 2 - 2 x^{2} \). Find \( \displaystyle \dfrac{dy}{dx} \) using \( \displaystyle \dfrac{dy}{dx} = \dfrac{dy}{du}\dfrac{du}{dx} \).
  1. \[ \frac{d}{d u} \left(u^{2} + u\right) = 2 u + 1 \]
    dy/du.✓ Proved
  2. \[ \frac{d}{d x} \left(2 - 2 x^{2}\right) = - 4 x \]
    du/dx.✓ Proved
  3. \[ - 4 x \left. 2 u + 1 \right|_{\substack{ u=2 - 2 x^{2} }} = 16 x^{3} - 20 x \]
    Multiply, and write u in terms of x.✓ Proved
Answer \( \frac{dy}{dx} = 16 x^{3} - 20 x \)

✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 2.2219

easy
Let \( \displaystyle y = \frac{1}{u} \) and \( \displaystyle u = e^{- x} \). Find \( \displaystyle \dfrac{dy}{dx} \) using \( \displaystyle \dfrac{dy}{dx} = \dfrac{dy}{du}\dfrac{du}{dx} \).
  1. \[ \frac{d}{d u} \frac{1}{u} = - \frac{1}{u^{2}} \]
    dy/du.✓ Proved
  2. \[ \frac{d}{d x} e^{- x} = - e^{- x} \]
    du/dx.✓ Proved
  3. \[ - e^{- x} \left. - \frac{1}{u^{2}} \right|_{\substack{ u=e^{- x} }} = e^{x} \]
    Multiply, and write u in terms of x.✓ Proved
Answer \( \frac{dy}{dx} = e^{x} \)

✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 2.2220

easy
Let \( \displaystyle y = e^{u} \) and \( \displaystyle u = e^{x} \). Find \( \displaystyle \dfrac{dy}{dx} \) using \( \displaystyle \dfrac{dy}{dx} = \dfrac{dy}{du}\dfrac{du}{dx} \).
  1. \[ \frac{d}{d u} e^{u} = e^{u} \]
    dy/du.✓ Proved
  2. \[ \frac{d}{d x} e^{x} = e^{x} \]
    du/dx.✓ Proved
  3. \[ e^{x} \left. e^{u} \right|_{\substack{ u=e^{x} }} = e^{x + e^{x}} \]
    Multiply, and write u in terms of x.✓ Proved
Answer \( \frac{dy}{dx} = e^{x + e^{x}} \)

✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 2.2221

easy
Let \( \displaystyle y = \sqrt{u} \) and \( \displaystyle u = - 2 x^{2} - 3 \). Find \( \displaystyle \dfrac{dy}{dx} \) using \( \displaystyle \dfrac{dy}{dx} = \dfrac{dy}{du}\dfrac{du}{dx} \).
  1. \[ \frac{d}{d u} \sqrt{u} = \frac{1}{2 \sqrt{u}} \]
    dy/du.✓ Proved
  2. \[ \frac{d}{d x} \left(- 2 x^{2} - 3\right) = - 4 x \]
    du/dx.✓ Proved
  3. \[ - 4 x \left. 1 \frac{1}{2 \sqrt{u}} \right|_{\substack{ u=- 2 x^{2} - 3 }} = \frac{2 i x}{\sqrt{2 x^{2} + 3}} \]
    Multiply, and write u in terms of x.✓ Proved
Answer \( \frac{dy}{dx} = \frac{2 i x}{\sqrt{2 x^{2} + 3}} \)

✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 2.2222

easy
Let \( \displaystyle y = e^{u} \) and \( \displaystyle u = x^{3} - x \). Find \( \displaystyle \dfrac{dy}{dx} \) using \( \displaystyle \dfrac{dy}{dx} = \dfrac{dy}{du}\dfrac{du}{dx} \).
  1. \[ \frac{d}{d u} e^{u} = e^{u} \]
    dy/du.✓ Proved
  2. \[ \frac{d}{d x} \left(x^{3} - x\right) = 3 x^{2} - 1 \]
    du/dx.✓ Proved
  3. \[ \left(3 x^{2} - 1\right) \left. e^{u} \right|_{\substack{ u=x^{3} - x }} = \left(3 x^{2} - 1\right) e^{x \left(x^{2} - 1\right)} \]
    Multiply, and write u in terms of x.✓ Proved
Answer \( \frac{dy}{dx} = \left(3 x^{2} - 1\right) e^{x \left(x^{2} - 1\right)} \)

✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 2.2223

easy
Let \( \displaystyle y = \sin{\left(u \right)} \) and \( \displaystyle u = 2 x + 2 \). Find \( \displaystyle \dfrac{dy}{dx} \) using \( \displaystyle \dfrac{dy}{dx} = \dfrac{dy}{du}\dfrac{du}{dx} \).
  1. \[ \frac{d}{d u} \sin{\left(u \right)} = \cos{\left(u \right)} \]
    dy/du.✓ Proved
  2. \[ \frac{d}{d x} \left(2 x + 2\right) = 2 \]
    du/dx.✓ Proved
  3. \[ \left. \cos{\left(u \right)} \right|_{\substack{ u=2 x + 2 }} \cdot 2 = 2 \cos{\left(2 x + 2 \right)} \]
    Multiply, and write u in terms of x.✓ Proved
Answer \( \frac{dy}{dx} = 2 \cos{\left(2 x + 2 \right)} \)

✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 2.2224

easy
Let \( \displaystyle y = \frac{1}{u} \) and \( \displaystyle u = x^{3} - x \). Find \( \displaystyle \dfrac{dy}{dx} \) using \( \displaystyle \dfrac{dy}{dx} = \dfrac{dy}{du}\dfrac{du}{dx} \).
  1. \[ \frac{d}{d u} \frac{1}{u} = - \frac{1}{u^{2}} \]
    dy/du.✓ Proved
  2. \[ \frac{d}{d x} \left(x^{3} - x\right) = 3 x^{2} - 1 \]
    du/dx.✓ Proved
  3. \[ \left(3 x^{2} - 1\right) \left. - \frac{1}{u^{2}} \right|_{\substack{ u=x^{3} - x }} = \frac{1 - 3 x^{2}}{x^{2} \left(x^{2} - 1\right)^{2}} \]
    Multiply, and write u in terms of x.✓ Proved
Answer \( \frac{dy}{dx} = \frac{1 - 3 x^{2}}{x^{2} \left(x^{2} - 1\right)^{2}} \)

✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 2.2225

easy
Let \( \displaystyle y = \frac{1}{u} \) and \( \displaystyle u = x^{2} + 1 \). Find \( \displaystyle \dfrac{dy}{dx} \) using \( \displaystyle \dfrac{dy}{dx} = \dfrac{dy}{du}\dfrac{du}{dx} \).
  1. \[ \frac{d}{d u} \frac{1}{u} = - \frac{1}{u^{2}} \]
    dy/du.✓ Proved
  2. \[ \frac{d}{d x} \left(x^{2} + 1\right) = 2 x \]
    du/dx.✓ Proved
  3. \[ 2 x \left. - \frac{1}{u^{2}} \right|_{\substack{ u=x^{2} + 1 }} = - \frac{2 x}{\left(x^{2} + 1\right)^{2}} \]
    Multiply, and write u in terms of x.✓ Proved
Answer \( \frac{dy}{dx} = - \frac{2 x}{\left(x^{2} + 1\right)^{2}} \)

✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 2.2226

easy
Let \( \displaystyle y = u^{3} \) and \( \displaystyle u = \cos{\left(x \right)} \). Find \( \displaystyle \dfrac{dy}{dx} \) using \( \displaystyle \dfrac{dy}{dx} = \dfrac{dy}{du}\dfrac{du}{dx} \).
  1. \[ \frac{d}{d u} u^{3} = 3 u^{2} \]
    dy/du.✓ Proved
  2. \[ \frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)} \]
    du/dx.✓ Proved
  3. \[ - \sin{\left(x \right)} \left. 3 u^{2} \right|_{\substack{ u=\cos{\left(x \right)} }} = - 3 \sin{\left(x \right)} \cos^{2}{\left(x \right)} \]
    Multiply, and write u in terms of x.✓ Proved
Answer \( \frac{dy}{dx} = - 3 \sin{\left(x \right)} \cos^{2}{\left(x \right)} \)

✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 2.2227

easy
Let \( \displaystyle y = \frac{1}{u} \) and \( \displaystyle u = 2 x + 4 \). Find \( \displaystyle \dfrac{dy}{dx} \) using \( \displaystyle \dfrac{dy}{dx} = \dfrac{dy}{du}\dfrac{du}{dx} \).
  1. \[ \frac{d}{d u} \frac{1}{u} = - \frac{1}{u^{2}} \]
    dy/du.✓ Proved
  2. \[ \frac{d}{d x} \left(2 x + 4\right) = 2 \]
    du/dx.✓ Proved
  3. \[ \left. - \frac{1}{u^{2}} \right|_{\substack{ u=2 x + 4 }} \cdot 2 = - \frac{1}{2 \left(x + 2\right)^{2}} \]
    Multiply, and write u in terms of x.✓ Proved
Answer \( \frac{dy}{dx} = - \frac{1}{2 \left(x + 2\right)^{2}} \)

✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 2.2228

easy
Let \( \displaystyle y = \ln{\left(u \right)} \) and \( \displaystyle u = 3 - 3 x^{2} \). Find \( \displaystyle \dfrac{dy}{dx} \) using \( \displaystyle \dfrac{dy}{dx} = \dfrac{dy}{du}\dfrac{du}{dx} \).
  1. \[ \frac{d}{d u} \ln{\left(u \right)} = \frac{1}{u} \]
    dy/du.✓ Proved
  2. \[ \frac{d}{d x} \left(3 - 3 x^{2}\right) = - 6 x \]
    du/dx.✓ Proved
  3. \[ - 6 x \left. 1 \frac{1}{u} \right|_{\substack{ u=3 - 3 x^{2} }} = \frac{2 x}{x^{2} - 1} \]
    Multiply, and write u in terms of x.✓ Proved
Answer \( \frac{dy}{dx} = \frac{2 x}{x^{2} - 1} \)

✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 2.2229

easy
Let \( \displaystyle y = \sin{\left(u \right)} \) and \( \displaystyle u = \cos{\left(x \right)} \). Find \( \displaystyle \dfrac{dy}{dx} \) using \( \displaystyle \dfrac{dy}{dx} = \dfrac{dy}{du}\dfrac{du}{dx} \).
  1. \[ \frac{d}{d u} \sin{\left(u \right)} = \cos{\left(u \right)} \]
    dy/du.✓ Proved
  2. \[ \frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)} \]
    du/dx.✓ Proved
  3. \[ - \sin{\left(x \right)} \left. \cos{\left(u \right)} \right|_{\substack{ u=\cos{\left(x \right)} }} = - \sin{\left(x \right)} \cos{\left(\cos{\left(x \right)} \right)} \]
    Multiply, and write u in terms of x.✓ Proved
Answer \( \frac{dy}{dx} = - \sin{\left(x \right)} \cos{\left(\cos{\left(x \right)} \right)} \)

✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 2.2230

easy
Let \( \displaystyle y = \ln{\left(u \right)} \) and \( \displaystyle u = 2 x - 2 \). Find \( \displaystyle \dfrac{dy}{dx} \) using \( \displaystyle \dfrac{dy}{dx} = \dfrac{dy}{du}\dfrac{du}{dx} \).
  1. \[ \frac{d}{d u} \ln{\left(u \right)} = \frac{1}{u} \]
    dy/du.✓ Proved
  2. \[ \frac{d}{d x} \left(2 x - 2\right) = 2 \]
    du/dx.✓ Proved
  3. \[ \left. 1 \frac{1}{u} \right|_{\substack{ u=2 x - 2 }} \cdot 2 = \frac{1}{x - 1} \]
    Multiply, and write u in terms of x.✓ Proved
Answer \( \frac{dy}{dx} = \frac{1}{x - 1} \)

✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 2.2231

easy
Let \( \displaystyle y = \frac{1}{u} \) and \( \displaystyle u = \cos{\left(x \right)} \). Find \( \displaystyle \dfrac{dy}{dx} \) using \( \displaystyle \dfrac{dy}{dx} = \dfrac{dy}{du}\dfrac{du}{dx} \).
  1. \[ \frac{d}{d u} \frac{1}{u} = - \frac{1}{u^{2}} \]
    dy/du.✓ Proved
  2. \[ \frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)} \]
    du/dx.✓ Proved
  3. \[ - \sin{\left(x \right)} \left. - \frac{1}{u^{2}} \right|_{\substack{ u=\cos{\left(x \right)} }} = \frac{\sin{\left(x \right)}}{\cos^{2}{\left(x \right)}} \]
    Multiply, and write u in terms of x.✓ Proved
Answer \( \frac{dy}{dx} = \frac{\sin{\left(x \right)}}{\cos^{2}{\left(x \right)}} \)

✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 2.2302

easy
Let \( \displaystyle y = e^{u} \) and \( \displaystyle u = 2 x^{2} + 3 \). Find \( \displaystyle \dfrac{dy}{dx} \) using \( \displaystyle \dfrac{dy}{dx} = \dfrac{dy}{du}\dfrac{du}{dx} \).
  1. \[ \frac{d}{d u} e^{u} = e^{u} \]
    dy/du.✓ Proved
  2. \[ \frac{d}{d x} \left(2 x^{2} + 3\right) = 4 x \]
    du/dx.✓ Proved
  3. \[ 4 x \left. e^{u} \right|_{\substack{ u=2 x^{2} + 3 }} = 4 x e^{2 x^{2} + 3} \]
    Multiply, and write u in terms of x.✓ Proved
Answer \( \frac{dy}{dx} = 4 x e^{2 x^{2} + 3} \)

✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 2.2305

easy
Let \( \displaystyle y = u^{2} + 3 u \) and \( \displaystyle u = e^{- x} \). Find \( \displaystyle \dfrac{dy}{dx} \) using \( \displaystyle \dfrac{dy}{dx} = \dfrac{dy}{du}\dfrac{du}{dx} \).
  1. \[ \frac{d}{d u} \left(u^{2} + 3 u\right) = 2 u + 3 \]
    dy/du.✓ Proved
  2. \[ \frac{d}{d x} e^{- x} = - e^{- x} \]
    du/dx.✓ Proved
  3. \[ - e^{- x} \left. 2 u + 3 \right|_{\substack{ u=e^{- x} }} = \left(- 3 e^{x} - 2\right) e^{- 2 x} \]
    Multiply, and write u in terms of x.✓ Proved
Answer \( \frac{dy}{dx} = \left(- 3 e^{x} - 2\right) e^{- 2 x} \)

✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 2.2306

easy
Let \( \displaystyle y = \sin{\left(u \right)} \) and \( \displaystyle u = 4 - x^{2} \). Find \( \displaystyle \dfrac{dy}{dx} \) using \( \displaystyle \dfrac{dy}{dx} = \dfrac{dy}{du}\dfrac{du}{dx} \).
  1. \[ \frac{d}{d u} \sin{\left(u \right)} = \cos{\left(u \right)} \]
    dy/du.✓ Proved
  2. \[ \frac{d}{d x} \left(4 - x^{2}\right) = - 2 x \]
    du/dx.✓ Proved
  3. \[ - 2 x \left. \cos{\left(u \right)} \right|_{\substack{ u=4 - x^{2} }} = - 2 x \cos{\left(x^{2} - 4 \right)} \]
    Multiply, and write u in terms of x.✓ Proved
Answer \( \frac{dy}{dx} = - 2 x \cos{\left(x^{2} - 4 \right)} \)

✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 2.2307

easy
Let \( \displaystyle y = e^{u} \) and \( \displaystyle u = \cos{\left(x \right)} \). Find \( \displaystyle \dfrac{dy}{dx} \) using \( \displaystyle \dfrac{dy}{dx} = \dfrac{dy}{du}\dfrac{du}{dx} \).
  1. \[ \frac{d}{d u} e^{u} = e^{u} \]
    dy/du.✓ Proved
  2. \[ \frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)} \]
    du/dx.✓ Proved
  3. \[ - \sin{\left(x \right)} \left. e^{u} \right|_{\substack{ u=\cos{\left(x \right)} }} = - e^{\cos{\left(x \right)}} \sin{\left(x \right)} \]
    Multiply, and write u in terms of x.✓ Proved
Answer \( \frac{dy}{dx} = - e^{\cos{\left(x \right)}} \sin{\left(x \right)} \)

✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 2.2308

easy
Let \( \displaystyle y = \sqrt{u} \) and \( \displaystyle u = e^{- 2 x} \). Find \( \displaystyle \dfrac{dy}{dx} \) using \( \displaystyle \dfrac{dy}{dx} = \dfrac{dy}{du}\dfrac{du}{dx} \).
  1. \[ \frac{d}{d u} \sqrt{u} = \frac{1}{2 \sqrt{u}} \]
    dy/du.✓ Proved
  2. \[ \frac{d}{d x} e^{- 2 x} = - 2 e^{- 2 x} \]
    du/dx.✓ Proved
  3. \[ - 2 e^{- 2 x} \left. 1 \frac{1}{2 \sqrt{u}} \right|_{\substack{ u=e^{- 2 x} }} = - e^{- x} \]
    Multiply, and write u in terms of x.✓ Proved
Answer \( \frac{dy}{dx} = - e^{- x} \)

✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 2.2309

easy
Let \( \displaystyle y = e^{u} \) and \( \displaystyle u = x^{2} + 1 \). Find \( \displaystyle \dfrac{dy}{dx} \) using \( \displaystyle \dfrac{dy}{dx} = \dfrac{dy}{du}\dfrac{du}{dx} \).
  1. \[ \frac{d}{d u} e^{u} = e^{u} \]
    dy/du.✓ Proved
  2. \[ \frac{d}{d x} \left(x^{2} + 1\right) = 2 x \]
    du/dx.✓ Proved
  3. \[ 2 x \left. e^{u} \right|_{\substack{ u=x^{2} + 1 }} = 2 x e^{x^{2} + 1} \]
    Multiply, and write u in terms of x.✓ Proved
Answer \( \frac{dy}{dx} = 2 x e^{x^{2} + 1} \)

✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

More the chain rule in leibniz notation practice

Let \( \displaystyle y = \ln{\left(u \right)} \) and \( \displaystyle u = \cos{\left(x \right)} \). Find \( \displaystyle \dfrac{dy}{dx} \) using \( \displaystyle \dfrac{dy}{dx} = \dfrac{dy}{du}\dfrac{du}{dx} \).
Problem 2.2218easy✓ Every equation proved
Let \( \displaystyle y = \frac{1}{u} \) and \( \displaystyle u = x^{2} + 4 \). Find \( \displaystyle \dfrac{dy}{dx} \) using \( \displaystyle \dfrac{dy}{dx} = \dfrac{dy}{du}\dfrac{du}{dx} \).
Problem 2.2301easy✓ Nihil obstat
Let \( \displaystyle y = - 2 u - 4 \) and \( \displaystyle u = - 3 x^{2} - 1 \). Find \( \displaystyle \dfrac{dy}{dx} \) using \( \displaystyle \dfrac{dy}{dx} = \dfrac{dy}{du}\dfrac{du}{dx} \).
Problem 2.2303easy✓ Every equation proved
Let \( \displaystyle y = - 4 u - 4 \) and \( \displaystyle u = x^{2} + 1 \). Find \( \displaystyle \dfrac{dy}{dx} \) using \( \displaystyle \dfrac{dy}{dx} = \dfrac{dy}{du}\dfrac{du}{dx} \).
Problem 2.2304easy✓ Every equation proved
Let \( \displaystyle y = \sin{\left(u \right)} \) and \( \displaystyle u = 2 x + 4 \). Find \( \displaystyle \dfrac{dy}{dx} \) using \( \displaystyle \dfrac{dy}{dx} = \dfrac{dy}{du}\dfrac{du}{dx} \).
Problem 2.2310easy✓ Nihil obstat
Let \( \displaystyle y = u^{3} \) and \( \displaystyle u = x^{3} - x \). Find \( \displaystyle \dfrac{dy}{dx} \) using \( \displaystyle \dfrac{dy}{dx} = \dfrac{dy}{du}\dfrac{du}{dx} \).
Problem 2.2331easy✓ Every equation proved
Let \( \displaystyle y = \sqrt{u} \) and \( \displaystyle u = \cos{\left(x \right)} \). Find \( \displaystyle \dfrac{dy}{dx} \) using \( \displaystyle \dfrac{dy}{dx} = \dfrac{dy}{du}\dfrac{du}{dx} \).
Problem 2.2332easy✓ Nihil obstat
Let \( \displaystyle y = \frac{1}{u} \) and \( \displaystyle u = 2 - x^{2} \). Find \( \displaystyle \dfrac{dy}{dx} \) using \( \displaystyle \dfrac{dy}{dx} = \dfrac{dy}{du}\dfrac{du}{dx} \).
Problem 2.2333easy✓ Nihil obstat
Let \( \displaystyle y = u^{4} \) and \( \displaystyle u = 2 x + 5 \). Find \( \displaystyle \dfrac{dy}{dx} \) using \( \displaystyle \dfrac{dy}{dx} = \dfrac{dy}{du}\dfrac{du}{dx} \).
Problem 2.2334easy✓ Nihil obstat
Let \( \displaystyle y = e^{u} \) and \( \displaystyle u = 2 x^{2} - 1 \). Find \( \displaystyle \dfrac{dy}{dx} \) using \( \displaystyle \dfrac{dy}{dx} = \dfrac{dy}{du}\dfrac{du}{dx} \).
Problem 2.2335easy✓ Nihil obstat
Let \( \displaystyle y = \sqrt{u} \) and \( \displaystyle u = 4 - 2 x^{2} \). Find \( \displaystyle \dfrac{dy}{dx} \) using \( \displaystyle \dfrac{dy}{dx} = \dfrac{dy}{du}\dfrac{du}{dx} \).
Problem 2.2336easy✓ Nihil obstat
Let \( \displaystyle y = u^{2} + u \) and \( \displaystyle u = 2 x - 5 \). Find \( \displaystyle \dfrac{dy}{dx} \) using \( \displaystyle \dfrac{dy}{dx} = \dfrac{dy}{du}\dfrac{du}{dx} \).
Problem 2.2337easy✓ Nihil obstat
Let \( \displaystyle y = 4 u + 2 \) and \( \displaystyle u = e^{2 x} \). Find \( \displaystyle \dfrac{dy}{dx} \) using \( \displaystyle \dfrac{dy}{dx} = \dfrac{dy}{du}\dfrac{du}{dx} \).
Problem 2.2338easy✓ Every equation proved
Let \( \displaystyle y = \sin{\left(u \right)} \) and \( \displaystyle u = x^{2} + 1 \). Find \( \displaystyle \dfrac{dy}{dx} \) using \( \displaystyle \dfrac{dy}{dx} = \dfrac{dy}{du}\dfrac{du}{dx} \).
Problem 2.2339easy✓ Nihil obstat
Let \( \displaystyle y = \ln{\left(u \right)} \) and \( \displaystyle u = e^{- 2 x} \). Find \( \displaystyle \dfrac{dy}{dx} \) using \( \displaystyle \dfrac{dy}{dx} = \dfrac{dy}{du}\dfrac{du}{dx} \).
Problem 2.2340easy✓ Nihil obstat
Let \( \displaystyle y = \ln{\left(u \right)} \) and \( \displaystyle u = x^{3} - x \). Find \( \displaystyle \dfrac{dy}{dx} \) using \( \displaystyle \dfrac{dy}{dx} = \dfrac{dy}{du}\dfrac{du}{dx} \).
Problem 2.2341easy✓ Nihil obstat
Let \( \displaystyle y = u^{4} \) and \( \displaystyle u = x^{3} - x \). Find \( \displaystyle \dfrac{dy}{dx} \) using \( \displaystyle \dfrac{dy}{dx} = \dfrac{dy}{du}\dfrac{du}{dx} \).
Problem 2.2342easy✓ Nihil obstat
Let \( \displaystyle y = \sin{\left(u \right)} \) and \( \displaystyle u = x^{3} - x \). Find \( \displaystyle \dfrac{dy}{dx} \) using \( \displaystyle \dfrac{dy}{dx} = \dfrac{dy}{du}\dfrac{du}{dx} \).
Problem 2.2343easy✓ Nihil obstat
Let \( \displaystyle y = \sqrt{u} \) and \( \displaystyle u = x^{2} + 1 \). Find \( \displaystyle \dfrac{dy}{dx} \) using \( \displaystyle \dfrac{dy}{dx} = \dfrac{dy}{du}\dfrac{du}{dx} \).
Problem 2.2344easy✓ Nihil obstat
Let \( \displaystyle y = \sin{\left(u \right)} \) and \( \displaystyle u = - 2 x^{2} - 4 \). Find \( \displaystyle \dfrac{dy}{dx} \) using \( \displaystyle \dfrac{dy}{dx} = \dfrac{dy}{du}\dfrac{du}{dx} \).
Problem 2.2345easy✓ Every equation proved
Let \( \displaystyle y = u^{2} + 2 u \) and \( \displaystyle u = x^{2} + 4 \). Find \( \displaystyle \dfrac{dy}{dx} \) using \( \displaystyle \dfrac{dy}{dx} = \dfrac{dy}{du}\dfrac{du}{dx} \).
Problem 2.2429easy✓ Nihil obstat
Let \( \displaystyle y = u^{5} \) and \( \displaystyle u = x^{2} + 1 \). Find \( \displaystyle \dfrac{dy}{dx} \) using \( \displaystyle \dfrac{dy}{dx} = \dfrac{dy}{du}\dfrac{du}{dx} \).
Problem 2.2430easy✓ Nihil obstat
Let \( \displaystyle y = \sqrt{u} \) and \( \displaystyle u = 2 x - 4 \). Find \( \displaystyle \dfrac{dy}{dx} \) using \( \displaystyle \dfrac{dy}{dx} = \dfrac{dy}{du}\dfrac{du}{dx} \).
Problem 2.2431easy✓ Nihil obstat
Let \( \displaystyle y = 3 - 4 u \) and \( \displaystyle u = x^{2} + 1 \). Find \( \displaystyle \dfrac{dy}{dx} \) using \( \displaystyle \dfrac{dy}{dx} = \dfrac{dy}{du}\dfrac{du}{dx} \).
Problem 2.2432easy✓ Nihil obstat
Let \( \displaystyle y = \ln{\left(u \right)} \) and \( \displaystyle u = e^{- x} \). Find \( \displaystyle \dfrac{dy}{dx} \) using \( \displaystyle \dfrac{dy}{dx} = \dfrac{dy}{du}\dfrac{du}{dx} \).
Problem 2.2433easy✓ Nihil obstat
Let \( \displaystyle y = u^{4} \) and \( \displaystyle u = \cos{\left(x \right)} \). Find \( \displaystyle \dfrac{dy}{dx} \) using \( \displaystyle \dfrac{dy}{dx} = \dfrac{dy}{du}\dfrac{du}{dx} \).
Problem 2.2434easy✓ Nihil obstat
Let \( \displaystyle y = u^{2} + u \) and \( \displaystyle u = e^{2 x} \). Find \( \displaystyle \dfrac{dy}{dx} \) using \( \displaystyle \dfrac{dy}{dx} = \dfrac{dy}{du}\dfrac{du}{dx} \).
Problem 2.2435easy✓ Nihil obstat
Let \( \displaystyle y = u^{2} + u \) and \( \displaystyle u = x^{3} - x \). Find \( \displaystyle \dfrac{dy}{dx} \) using \( \displaystyle \dfrac{dy}{dx} = \dfrac{dy}{du}\dfrac{du}{dx} \).
Problem 2.2436easy✓ Nihil obstat
Let \( \displaystyle y = u^{2} - u \) and \( \displaystyle u = \cos{\left(x \right)} \). Find \( \displaystyle \dfrac{dy}{dx} \) using \( \displaystyle \dfrac{dy}{dx} = \dfrac{dy}{du}\dfrac{du}{dx} \).
Problem 2.2437easy✓ Every equation proved
Let \( \displaystyle y = \ln{\left(u \right)} \) and \( \displaystyle u = x^{2} + 1 \). Find \( \displaystyle \dfrac{dy}{dx} \) using \( \displaystyle \dfrac{dy}{dx} = \dfrac{dy}{du}\dfrac{du}{dx} \).
Problem 2.2438easy✓ Nihil obstat
Let \( \displaystyle y = - 2 u - 2 \) and \( \displaystyle u = 2 x - 4 \). Find \( \displaystyle \dfrac{dy}{dx} \) using \( \displaystyle \dfrac{dy}{dx} = \dfrac{dy}{du}\dfrac{du}{dx} \).
Problem 2.2479easy✓ Nihil obstat
Let \( \displaystyle y = u^{2} + 3 u \) and \( \displaystyle u = x^{3} - x \). Find \( \displaystyle \dfrac{dy}{dx} \) using \( \displaystyle \dfrac{dy}{dx} = \dfrac{dy}{du}\dfrac{du}{dx} \).
Problem 2.2480easy✓ Nihil obstat
Let \( \displaystyle y = \sin{\left(u \right)} \) and \( \displaystyle u = e^{- x} \). Find \( \displaystyle \dfrac{dy}{dx} \) using \( \displaystyle \dfrac{dy}{dx} = \dfrac{dy}{du}\dfrac{du}{dx} \).
Problem 2.2481easy✓ Nihil obstat
Let \( \displaystyle y = u^{2} + 2 u \) and \( \displaystyle u = \cos{\left(x \right)} \). Find \( \displaystyle \dfrac{dy}{dx} \) using \( \displaystyle \dfrac{dy}{dx} = \dfrac{dy}{du}\dfrac{du}{dx} \).
Problem 2.2482easy✓ Nihil obstat
Let \( \displaystyle y = e^{u} \) and \( \displaystyle u = 4 - 3 x^{2} \). Find \( \displaystyle \dfrac{dy}{dx} \) using \( \displaystyle \dfrac{dy}{dx} = \dfrac{dy}{du}\dfrac{du}{dx} \).
Problem 2.2483easy✓ Nihil obstat
Let \( \displaystyle y = - 4 u - 2 \) and \( \displaystyle u = 2 - 2 x^{2} \). Find \( \displaystyle \dfrac{dy}{dx} \) using \( \displaystyle \dfrac{dy}{dx} = \dfrac{dy}{du}\dfrac{du}{dx} \).
Problem 2.2484easy✓ Every equation proved
Let \( \displaystyle y = \sqrt{u} \) and \( \displaystyle u = 2 - x^{2} \). Find \( \displaystyle \dfrac{dy}{dx} \) using \( \displaystyle \dfrac{dy}{dx} = \dfrac{dy}{du}\dfrac{du}{dx} \).
Problem 2.2485easy✓ Nihil obstat
Let \( \displaystyle y = 4 u - 5 \) and \( \displaystyle u = e^{- x} \). Find \( \displaystyle \dfrac{dy}{dx} \) using \( \displaystyle \dfrac{dy}{dx} = \dfrac{dy}{du}\dfrac{du}{dx} \).
Problem 2.2486easy✓ Nihil obstat
Let \( \displaystyle y = \sqrt{u} \) and \( \displaystyle u = - 2 x^{2} - 4 \). Find \( \displaystyle \dfrac{dy}{dx} \) using \( \displaystyle \dfrac{dy}{dx} = \dfrac{dy}{du}\dfrac{du}{dx} \).
Problem 2.2487easy✓ Nihil obstat
Let \( \displaystyle y = u^{3} \) and \( \displaystyle u = x^{2} + 1 \). Find \( \displaystyle \dfrac{dy}{dx} \) using \( \displaystyle \dfrac{dy}{dx} = \dfrac{dy}{du}\dfrac{du}{dx} \).
Problem 2.2488easy✓ Nihil obstat