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Second derivatives practice problems

Differentiate twice: concavity, acceleration, and higher-order derivatives. 30 problems with worked solutions; in 30 of them every equation is proved by a computer algebra system.

Find \( \displaystyle \dfrac{d^2y}{dx^2} \) for \( \displaystyle y = \sqrt{x^{2} + 4 x + 4} \).
Problem 2.790easy✓ Every equation proved
Find \( \displaystyle \dfrac{d^2y}{dx^2} \) for \( \displaystyle y = x^{3} - 2 \cos{\left(x \right)} \).
Problem 2.740medium✓ Every equation proved
Find \( \displaystyle \dfrac{d^2y}{dx^2} \) for \( \displaystyle y = \ln{\left(x^{2} + 3 \right)} \).
Problem 2.741medium✓ Every equation proved
Find \( \displaystyle \dfrac{d^2y}{dx^2} \) for \( \displaystyle y = x \sin{\left(x \right)} \).
Problem 2.742medium✓ Every equation proved
Find \( \displaystyle \dfrac{d^2y}{dx^2} \) for \( \displaystyle y = \ln{\left(x^{2} + 4 \right)} \).
Problem 2.743medium✓ Every equation proved
Find \( \displaystyle \dfrac{d^2y}{dx^2} \) for \( \displaystyle y = e^{x} \cos{\left(x \right)} \).
Problem 2.744medium✓ Every equation proved
Find \( \displaystyle \dfrac{d^2y}{dx^2} \) for \( \displaystyle y = x^{4} - 2 \cos{\left(x \right)} \).
Problem 2.745medium✓ Every equation proved
Find \( \displaystyle \dfrac{d^2y}{dx^2} \) for \( \displaystyle y = \sqrt{x^{2} + 2 x + 3} \).
Problem 2.747medium✓ Every equation proved
Find \( \displaystyle \dfrac{d^2y}{dx^2} \) for \( \displaystyle y = \ln{\left(x^{2} + 1 \right)} \).
Problem 2.748medium✓ Every equation proved
Find \( \displaystyle \dfrac{d^2y}{dx^2} \) for \( \displaystyle y = x^{2} e^{- x} \).
Problem 2.749medium✓ Every equation proved
Find \( \displaystyle \dfrac{d^2y}{dx^2} \) for \( \displaystyle y = \sqrt{x^{2} + x + 2} \).
Problem 2.750medium✓ Every equation proved
Find \( \displaystyle \dfrac{d^2y}{dx^2} \) for \( \displaystyle y = x^{3} - \cos{\left(x \right)} \).
Problem 2.751medium✓ Every equation proved
Find \( \displaystyle \dfrac{d^2y}{dx^2} \) for \( \displaystyle y = x^{2} e^{- 2 x} \).
Problem 2.752medium✓ Every equation proved
Find \( \displaystyle \dfrac{d^2y}{dx^2} \) for \( \displaystyle y = x^{2} e^{2 x} \).
Problem 2.753medium✓ Every equation proved
Find \( \displaystyle \dfrac{d^2y}{dx^2} \) for \( \displaystyle y = \sqrt{x^{2} + 2 x + 2} \).
Problem 2.754medium✓ Every equation proved
Find \( \displaystyle \dfrac{d^2y}{dx^2} \) for \( \displaystyle y = \sqrt{x^{2} + x + 1} \).
Problem 2.785medium✓ Every equation proved
Find \( \displaystyle \dfrac{d^2y}{dx^2} \) for \( \displaystyle y = x^{4} - 4 \cos{\left(x \right)} \).
Problem 2.786medium✓ Every equation proved
Find \( \displaystyle \dfrac{d^2y}{dx^2} \) for \( \displaystyle y = x^{2} e^{x} \).
Problem 2.787medium✓ Every equation proved
Find \( \displaystyle \dfrac{d^2y}{dx^2} \) for \( \displaystyle y = x^{5} - \cos{\left(x \right)} \).
Problem 2.788medium✓ Every equation proved
Find \( \displaystyle \dfrac{d^2y}{dx^2} \) for \( \displaystyle y = \sqrt{x^{2} + 4 x + 3} \).
Problem 2.789medium✓ Every equation proved
Find \( \displaystyle \dfrac{d^2y}{dx^2} \) for \( \displaystyle y = x^{5} - 3 \cos{\left(x \right)} \).
Problem 2.791medium✓ Every equation proved
Find \( \displaystyle \dfrac{d^2y}{dx^2} \) for \( \displaystyle y = \sqrt{x^{2} + 2 x + 4} \).
Problem 2.792medium✓ Every equation proved
Find \( \displaystyle \dfrac{d^2y}{dx^2} \) for \( \displaystyle y = x^{6} - 4 \cos{\left(x \right)} \).
Problem 2.793medium✓ Every equation proved
Find \( \displaystyle \dfrac{d^2y}{dx^2} \) for \( \displaystyle y = x^{4} - 5 \cos{\left(x \right)} \).
Problem 2.794medium✓ Every equation proved
Find \( \displaystyle \dfrac{d^2y}{dx^2} \) for \( \displaystyle y = x^{5} - 5 \cos{\left(x \right)} \).
Problem 2.795medium✓ Every equation proved
Find \( \displaystyle \dfrac{d^2y}{dx^2} \) for \( \displaystyle y = \sqrt{x^{2} + 4 x + 2} \).
Problem 2.796medium✓ Every equation proved
Find \( \displaystyle \dfrac{d^2y}{dx^2} \) for \( \displaystyle y = x^{6} - 5 \cos{\left(x \right)} \).
Problem 2.797medium✓ Every equation proved
Find \( \displaystyle \dfrac{d^2y}{dx^2} \) for \( \displaystyle y = x^{6} - 2 \cos{\left(x \right)} \).
Problem 2.798medium✓ Every equation proved
Find \( \displaystyle \dfrac{d^2y}{dx^2} \) for \( \displaystyle y = \sqrt{x^{2} + 3 x + 4} \).
Problem 2.799medium✓ Every equation proved
Find \( \displaystyle \dfrac{d^2y}{dx^2} \) for \( \displaystyle y = \frac{x}{x^{2} + 1} \).
Problem 2.746hard✓ Every equation proved