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:

• Derivatives of basic trigonometric functions ($\sin x$, $\cos x$, $\tan x$, $\cot x$, $\sec x$)

• Derivatives of inverse trigonometric functions (${\sin }^{-1}x$, ${\cos }^{-1}x$, ${\tan }^{-1}x$, ${\cot }^{-1}x$, ${\sec }^{-1}x$)

• Product rule for differentiation

• Quotient rule for differentiation

• Chain rule for composite functions

• Algebraic manipulation and simplification of trigonometric expressions

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Important Formulas

Below are the key formulas used in this exercise:

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

$\frac{d}{dx}(\sec x)=\sec x\tan x$

Inverse Trigonometric Derivatives: $\frac{d}{dx}({\sin }^{-1}x)=\frac{1}{\sqrt{1-x^2}},\frac{d}{dx}({\cos }^{-1}x)=-\frac{1}{\sqrt{1-x^2}}$

$\frac{d}{dx}({\tan }^{-1}x)=\frac{1}{1+x^2},\frac{d}{dx}({\cot }^{-1}x)=-\frac{1}{1+x^2}$ $\frac{d}{dx}({\sec }^{-1}x)=\frac{1}{|x|\sqrt{x^2-1}}$

Differentiation Rules: $\text{Product Rule: }\frac{d}{dx}(uv)=u\frac{dv}{dx}+v\frac{du}{dx}$

$\text{Quotient Rule: }\frac{d}{dx}\left(\frac{u}{v}\right)=\frac{v\frac{du}{dx}-u\frac{dv}{dx}}{v^2}$

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

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Summary

This exercise covers the differentiation of trigonometric and inverse trigonometric functions in various combinations. It progresses from basic polynomial-trig combinations to products and quotients requiring the product and quotient rules, then advances to inverse trigonometric functions requiring the chain rule. Key strategies include recognizing the appropriate differentiation rule based on function structure (sums, products, quotients, compositions), applying the chain rule carefully for composite inverse trig functions, and simplifying results using identities where applicable (notably ${\sin }^{-1}x+{\cos }^{-1}x=\frac{\pi }{2}$ in Q15).