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Second derivatives practice problems

Differentiate twice: concavity, acceleration, and higher-order derivatives. 47 problems with worked solutions; in 47 of them every equation is proved by a computer algebra system.

12 worked problems

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

Problem 2.746

hard
Find \( \displaystyle \dfrac{d^2y}{dx^2} \) for \( \displaystyle y = \frac{x}{x^{2} + 1} \).
  1. \[ \frac{d}{d x} \frac{x}{x^{2} + 1} = \frac{1 - x^{2}}{x^{4} + 2 x^{2} + 1} \]
    The first derivative.✓ Proved
  2. \[ \frac{d}{d x} \frac{1 - x^{2}}{x^{4} + 2 x^{2} + 1} = \frac{2 x \left(x^{2} - 3\right)}{x^{6} + 3 x^{4} + 3 x^{2} + 1} \]
    Differentiate again.✓ Proved
Answer \( \frac{2 x \left(x^{2} - 3\right)}{x^{6} + 3 x^{4} + 3 x^{2} + 1} \)

✓ 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 2.740

medium
Find \( \displaystyle \dfrac{d^2y}{dx^2} \) for \( \displaystyle y = x^{3} - 2 \cos{\left(x \right)} \).
  1. \[ \frac{d}{d x} \left(x^{3} - 2 \cos{\left(x \right)}\right) = 3 x^{2} + 2 \sin{\left(x \right)} \]
    The first derivative.✓ Proved
  2. \[ \frac{d}{d x} \left(3 x^{2} + 2 \sin{\left(x \right)}\right) = 6 x + 2 \cos{\left(x \right)} \]
    Differentiate again.✓ Proved
Answer \( 6 x + 2 \cos{\left(x \right)} \)

✓ 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 2.741

medium
Find \( \displaystyle \dfrac{d^2y}{dx^2} \) for \( \displaystyle y = \ln{\left(x^{2} + 3 \right)} \).
  1. \[ \frac{d}{d x} \ln{\left(x^{2} + 3 \right)} = \frac{2 x}{x^{2} + 3} \]
    The first derivative.✓ Proved
  2. \[ \frac{d}{d x} \frac{2 x}{x^{2} + 3} = \frac{6 - 2 x^{2}}{x^{4} + 6 x^{2} + 9} \]
    Differentiate again.✓ Proved
Answer \( \frac{2 \left(3 - x^{2}\right)}{x^{4} + 6 x^{2} + 9} \)

✓ 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 2.742

medium
Find \( \displaystyle \dfrac{d^2y}{dx^2} \) for \( \displaystyle y = x \sin{\left(x \right)} \).
  1. \[ \frac{d}{d x} x \sin{\left(x \right)} = x \cos{\left(x \right)} + \sin{\left(x \right)} \]
    The first derivative.✓ Proved
  2. \[ \frac{d}{d x} \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right) = - x \sin{\left(x \right)} + 2 \cos{\left(x \right)} \]
    Differentiate again.✓ Proved
Answer \( - x \sin{\left(x \right)} + 2 \cos{\left(x \right)} \)

✓ 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 2.744

medium
Find \( \displaystyle \dfrac{d^2y}{dx^2} \) for \( \displaystyle y = e^{x} \cos{\left(x \right)} \).
  1. \[ \frac{d}{d x} e^{x} \cos{\left(x \right)} = \sqrt{2} e^{x} \cos{\left(x + \frac{\pi}{4} \right)} \]
    The first derivative.✓ Proved
  2. \[ \frac{d}{d x} \sqrt{2} e^{x} \cos{\left(x + \frac{\pi}{4} \right)} = - 2 e^{x} \sin{\left(x \right)} \]
    Differentiate again.✓ Proved
Answer \( - 2 e^{x} \sin{\left(x \right)} \)

✓ 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 2.747

medium
Find \( \displaystyle \dfrac{d^2y}{dx^2} \) for \( \displaystyle y = \sqrt{x^{2} + 2 x + 3} \).
  1. \[ \frac{d}{d x} \sqrt{x^{2} + 2 x + 3} = \frac{x + 1}{\sqrt{x^{2} + 2 x + 3}} \]
    The first derivative.✓ Proved
  2. \[ \frac{d}{d x} \frac{x + 1}{\sqrt{x^{2} + 2 x + 3}} = \frac{2}{\left(x^{2} + 2 x + 3\right)^{\frac{3}{2}}} \]
    Differentiate again.✓ Proved
Answer \( \frac{2}{\left(x^{2} + 2 x + 3\right)^{\frac{3}{2}}} \)

✓ 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 2.749

medium
Find \( \displaystyle \dfrac{d^2y}{dx^2} \) for \( \displaystyle y = x^{2} e^{- x} \).
  1. \[ \frac{d}{d x} x^{2} e^{- x} = x \left(2 - x\right) e^{- x} \]
    The first derivative.✓ Proved
  2. \[ \frac{d}{d x} x \left(2 - x\right) e^{- x} = \left(x^{2} - 4 x + 2\right) e^{- x} \]
    Differentiate again.✓ Proved
Answer \( \left(x^{2} - 4 x + 2\right) e^{- x} \)

✓ 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 2.750

medium
Find \( \displaystyle \dfrac{d^2y}{dx^2} \) for \( \displaystyle y = \sqrt{x^{2} + x + 2} \).
  1. \[ \frac{d}{d x} \sqrt{x^{2} + x + 2} = \frac{x + \frac{1}{2}}{\sqrt{x^{2} + x + 2}} \]
    The first derivative.✓ Proved
  2. \[ \frac{d}{d x} \frac{x + \frac{1}{2}}{\sqrt{x^{2} + x + 2}} = \frac{7}{4 \left(x^{2} + x + 2\right)^{\frac{3}{2}}} \]
    Differentiate again.✓ Proved
Answer \( \frac{7}{4 \left(x^{2} + x + 2\right)^{\frac{3}{2}}} \)

✓ 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 2.751

medium
Find \( \displaystyle \dfrac{d^2y}{dx^2} \) for \( \displaystyle y = x^{3} - \cos{\left(x \right)} \).
  1. \[ \frac{d}{d x} \left(x^{3} - \cos{\left(x \right)}\right) = 3 x^{2} + \sin{\left(x \right)} \]
    The first derivative.✓ Proved
  2. \[ \frac{d}{d x} \left(3 x^{2} + \sin{\left(x \right)}\right) = 6 x + \cos{\left(x \right)} \]
    Differentiate again.✓ Proved
Answer \( 6 x + \cos{\left(x \right)} \)

✓ 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 2.752

medium
Find \( \displaystyle \dfrac{d^2y}{dx^2} \) for \( \displaystyle y = x^{2} e^{- 2 x} \).
  1. \[ \frac{d}{d x} x^{2} e^{- 2 x} = 2 x \left(1 - x\right) e^{- 2 x} \]
    The first derivative.✓ Proved
  2. \[ \frac{d}{d x} 2 x \left(1 - x\right) e^{- 2 x} = \left(4 x^{2} - 8 x + 2\right) e^{- 2 x} \]
    Differentiate again.✓ Proved
Answer \( 2 \left(2 x^{2} - 4 x + 1\right) e^{- 2 x} \)

✓ 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 2.753

medium
Find \( \displaystyle \dfrac{d^2y}{dx^2} \) for \( \displaystyle y = x^{2} e^{2 x} \).
  1. \[ \frac{d}{d x} x^{2} e^{2 x} = 2 x \left(x + 1\right) e^{2 x} \]
    The first derivative.✓ Proved
  2. \[ \frac{d}{d x} 2 x \left(x + 1\right) e^{2 x} = \left(4 x^{2} + 8 x + 2\right) e^{2 x} \]
    Differentiate again.✓ Proved
Answer \( \left(4 x^{2} + 8 x + 2\right) e^{2 x} \)

✓ 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 2.787

medium
Find \( \displaystyle \dfrac{d^2y}{dx^2} \) for \( \displaystyle y = x^{2} e^{x} \).
  1. \[ \frac{d}{d x} x^{2} e^{x} = x \left(x + 2\right) e^{x} \]
    The first derivative.✓ Proved
  2. \[ \frac{d}{d x} x \left(x + 2\right) e^{x} = \left(x^{2} + 4 x + 2\right) e^{x} \]
    Differentiate again.✓ Proved
Answer \( \left(x^{2} + 4 x + 2\right) e^{x} \)

✓ 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

More second derivatives practice

Find \( \displaystyle \dfrac{d^2y}{dx^2} \) for \( \displaystyle y = \sqrt{x^{2} + 4 x + 4} \).
Problem 2.790easy✓ Nihil obstat
Find \( \displaystyle \dfrac{d^2y}{dx^2} \) for \( \displaystyle y = \sqrt{x^{2} + 2 x + 1} \).
Problem 2.926easy✓ Every equation proved
Find \( \displaystyle \dfrac{d^2y}{dx^2} \) for \( \displaystyle y = \ln{\left(x^{2} + 4 \right)} \).
Problem 2.743medium✓ Nihil obstat
Find \( \displaystyle \dfrac{d^2y}{dx^2} \) for \( \displaystyle y = x^{4} - 2 \cos{\left(x \right)} \).
Problem 2.745medium✓ Nihil obstat
Find \( \displaystyle \dfrac{d^2y}{dx^2} \) for \( \displaystyle y = \ln{\left(x^{2} + 1 \right)} \).
Problem 2.748medium✓ Nihil obstat
Find \( \displaystyle \dfrac{d^2y}{dx^2} \) for \( \displaystyle y = \sqrt{x^{2} + 2 x + 2} \).
Problem 2.754medium✓ Nihil obstat
Find \( \displaystyle \dfrac{d^2y}{dx^2} \) for \( \displaystyle y = \sqrt{x^{2} + x + 1} \).
Problem 2.785medium✓ Nihil obstat
Find \( \displaystyle \dfrac{d^2y}{dx^2} \) for \( \displaystyle y = x^{4} - 4 \cos{\left(x \right)} \).
Problem 2.786medium✓ Nihil obstat
Find \( \displaystyle \dfrac{d^2y}{dx^2} \) for \( \displaystyle y = x^{5} - \cos{\left(x \right)} \).
Problem 2.788medium✓ Nihil obstat
Find \( \displaystyle \dfrac{d^2y}{dx^2} \) for \( \displaystyle y = \sqrt{x^{2} + 4 x + 3} \).
Problem 2.789medium✓ Nihil obstat
Find \( \displaystyle \dfrac{d^2y}{dx^2} \) for \( \displaystyle y = x^{5} - 3 \cos{\left(x \right)} \).
Problem 2.791medium✓ Nihil obstat
Find \( \displaystyle \dfrac{d^2y}{dx^2} \) for \( \displaystyle y = \sqrt{x^{2} + 2 x + 4} \).
Problem 2.792medium✓ Nihil obstat
Find \( \displaystyle \dfrac{d^2y}{dx^2} \) for \( \displaystyle y = x^{6} - 4 \cos{\left(x \right)} \).
Problem 2.793medium✓ Nihil obstat
Find \( \displaystyle \dfrac{d^2y}{dx^2} \) for \( \displaystyle y = x^{4} - 5 \cos{\left(x \right)} \).
Problem 2.794medium✓ Nihil obstat
Find \( \displaystyle \dfrac{d^2y}{dx^2} \) for \( \displaystyle y = x^{5} - 5 \cos{\left(x \right)} \).
Problem 2.795medium✓ Nihil obstat
Find \( \displaystyle \dfrac{d^2y}{dx^2} \) for \( \displaystyle y = \sqrt{x^{2} + 4 x + 2} \).
Problem 2.796medium✓ Nihil obstat
Find \( \displaystyle \dfrac{d^2y}{dx^2} \) for \( \displaystyle y = x^{6} - 5 \cos{\left(x \right)} \).
Problem 2.797medium✓ Nihil obstat
Find \( \displaystyle \dfrac{d^2y}{dx^2} \) for \( \displaystyle y = x^{6} - 2 \cos{\left(x \right)} \).
Problem 2.798medium✓ Nihil obstat
Find \( \displaystyle \dfrac{d^2y}{dx^2} \) for \( \displaystyle y = \sqrt{x^{2} + 3 x + 4} \).
Problem 2.799medium✓ Nihil obstat
Find \( \displaystyle \dfrac{d^2y}{dx^2} \) for \( \displaystyle y = x^{4} - 3 \cos{\left(x \right)} \).
Problem 2.919medium✓ Nihil obstat
Find \( \displaystyle \dfrac{d^2y}{dx^2} \) for \( \displaystyle y = x^{4} - \cos{\left(x \right)} \).
Problem 2.920medium✓ Nihil obstat
Find \( \displaystyle \dfrac{d^2y}{dx^2} \) for \( \displaystyle y = \ln{\left(x^{2} + 2 \right)} \).
Problem 2.921medium✓ Nihil obstat
Find \( \displaystyle \dfrac{d^2y}{dx^2} \) for \( \displaystyle y = x^{5} - 4 \cos{\left(x \right)} \).
Problem 2.922medium✓ Nihil obstat
Find \( \displaystyle \dfrac{d^2y}{dx^2} \) for \( \displaystyle y = \sqrt{x^{2} + 4 x + 1} \).
Problem 2.923medium✓ Nihil obstat
Find \( \displaystyle \dfrac{d^2y}{dx^2} \) for \( \displaystyle y = \sqrt{x^{2} + 3 x + 1} \).
Problem 2.924medium✓ Nihil obstat
Find \( \displaystyle \dfrac{d^2y}{dx^2} \) for \( \displaystyle y = x^{5} - 2 \cos{\left(x \right)} \).
Problem 2.925medium✓ Nihil obstat
Find \( \displaystyle \dfrac{d^2y}{dx^2} \) for \( \displaystyle y = x^{6} - \cos{\left(x \right)} \).
Problem 2.1044medium✓ Nihil obstat
Find \( \displaystyle \dfrac{d^2y}{dx^2} \) for \( \displaystyle y = \sqrt{x^{2} + x + 4} \).
Problem 2.1154medium✓ Nihil obstat
Find \( \displaystyle \dfrac{d^2y}{dx^2} \) for \( \displaystyle y = x^{3} - 4 \cos{\left(x \right)} \).
Problem 2.1155medium✓ Nihil obstat
Find \( \displaystyle \dfrac{d^2y}{dx^2} \) for \( \displaystyle y = x^{3} - 3 \cos{\left(x \right)} \).
Problem 2.1156medium✓ Nihil obstat
Find \( \displaystyle \dfrac{d^2y}{dx^2} \) for \( \displaystyle y = \sqrt{x^{2} + 3 x + 3} \).
Problem 2.1271medium✓ Nihil obstat
Find \( \displaystyle \dfrac{d^2y}{dx^2} \) for \( \displaystyle y = \sqrt{x^{2} + 3 x + 2} \).
Problem 2.1465medium✓ Nihil obstat
Find \( \displaystyle \dfrac{d^2y}{dx^2} \) for \( \displaystyle y = x^{3} - 5 \cos{\left(x \right)} \).
Problem 2.1466medium✓ Nihil obstat
Find \( \displaystyle \dfrac{d^2y}{dx^2} \) for \( \displaystyle y = \sqrt{x^{2} + x + 3} \).
Problem 2.1467medium✓ Nihil obstat
Find \( \displaystyle \dfrac{d^2y}{dx^2} \) for \( \displaystyle y = x^{6} - 3 \cos{\left(x \right)} \).
Problem 2.2532medium✓ Nihil obstat