Exercise Questions

All questions in this exercise are listed below. Click on a question to view its solution.

Key Concepts

This exercise focuses on the following concepts:

• Second derivatives and higher-order derivatives ($\frac{d^{n}y}{d{x}^n}$)

• Power Rule for differentiation: $\frac{d}{dx}(x^n)=n{x}^{n-1}$

• Chain Rule for composite functions

• Product Rule for differentiation of products

• Derivatives of trigonometric functions ($\sin$, $\cos$, $\tan$)

• Derivatives of exponential functions ($e^{ax}$)

• Derivatives of logarithmic functions ($\ln$)

• Leibniz Rule (generalized product rule for second derivatives)

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Important Formulas

Below are the key formulas used in this exercise:

Basic Differentiation Rules: $\frac{d}{dx}(x^n)=n{x}^{n-1}$

Higher-Order Derivative Notation: $f^{\prime \prime }(x)=\frac{d^{2}y}{d{x}^2},f^{(n)}(x)=\frac{d^{n}y}{d{x}^n}$

Chain Rule: $\frac{d}{dx}[f(g(x))]=f^{\prime }(g(x))\cdot g^{\prime }(x)$

Product Rule: $\frac{d}{dx}[f(x)g(x)]=f^{\prime }(x)g(x)+f(x)g^{\prime }(x)$

Leibniz Rule (Second Derivative of Product): $\frac{d^2}{d{x}^2}(fg)=f^{\prime \prime }g+2f^{\prime }{g}^{\prime }+f{g}^{\prime \prime }$

Trigonometric Derivatives: $\frac{d}{dx}(\sin x)=\cos x,\frac{d}{dx}(\cos x)=-\sin x,\frac{d}{dx}(\tan x)={\sec }^{2}x$

Exponential and Logarithmic Derivatives: $\frac{d}{dx}(e^{ax})=a{e}^{ax},\frac{d}{dx}(\ln x)=\frac{1}{x}$

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Summary

This exercise covers the computation of second and higher-order derivatives for a variety of function types including polynomials, rational functions, trigonometric, exponential, and logarithmic functions. The key strategy involves sequential application of differentiation rules—particularly the chain rule and product rule—to find successive derivatives.

Questions 15–18 extend to third, fourth, and fifth derivatives, requiring careful pattern recognition in repeated differentiation. Question 20 establishes the Leibniz rule, a generalization of the product rule for second derivatives that resembles the binomial expansion pattern.