05_Ex 2.5

Exercise Questions

Questions	Question Links
Q1. Differentiate trigonometric functions from first Principle.	2.5 Q-1
Q2. Differentiate the following w.r.t the variables involved.	2.5 Q-2
Q3. Find $\frac{d y}{d x}$ if,	2.5 Q-3
Q4. Differentiate $\cos x^2$ using the first principle.	2.5 Q-4
Q5. Differentiate.	2.5 Q-5
Q6. If $\tan y(1+\tan x)=1-\tan x$, show that $\frac{d y}{d x}=-1$	2.5 Q-6
Q7. $y=\sqrt{\sqrt{\tan x}+\sqrt{\tan x}+\sqrt{\tan x+} \ldots . \infty}$	2.5 Q-7
Q8. If $x = a \cos^{3} \theta$, $y = b \sin^{3} \theta$,	2.5 Q-8
Q9. Find $\frac{dy}{dx} \text{ if } x = a(\cos t + \sin t) \text{ and } y = a(\sin t - t \cos t)$	2.5 Q-9
Q10. Differentiate w.r.t ’ $x$ ‘	2.5 Q-10
Q11. if $\frac{y}{x}=\tan ^{-1}\left(\frac{y}{x}\right)$ show that $\frac{d y}{d x}=\frac{y}{x}$	2.5 Q-11
Q12. If $y=\tan \left(\tan ^{-1} x\right)$ show that $(1 + x^{2}) y_{1} - p(1 + y^{2}) = 0$	2.5 Q-12

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Overview

This exercise focuses on applying differentiation to trigonometric and inverse trigonometric functions. It explores basic principles like the first principle of differentiation and develops problem-solving techniques for various standard and complex forms of trigonometric functions. Understanding these concepts is crucial in calculus, particularly for solving problems in physics, engineering, and mathematics involving rates of change and oscillatory behavior.

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Key Concepts

• Trigonometric Derivatives:

  • $\frac{d}{dx} \sin x = \cos x$
  • $\frac{d}{dx} \cos x = -\sin x$
  • $\frac{d}{dx} \tan x = \sec^2 x$
  • Higher-order combinations like $\sin 2x$, $\tan^2x$, and $\cos x^2$.

• Inverse Trigonometric Derivatives:

  • $\frac{d}{dx} \sin^{-1}x = \frac{1}{\sqrt{1-x^2}}$
  • $\frac{d}{dx} \tan^{-1}x = \frac{1}{1+x^2}$

• Application of first principles for fundamental understanding of limits and differentiation.

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Important Formulas

1. Trigonometric Derivatives:

   • $\frac{d}{dx} \sin x = \cos x$
   • $\frac{d}{dx} \cos x = -\sin x$
   • $\frac{d}{dx} \tan x = \sec^2 x$
   • $\frac{d}{dx} \sec x = \sec x \tan x$
   • $\frac{d}{dx} \text{cosec}\space x = -\text{cosec}\space x \cot x$
   • $\frac{d}{dx} \cot x = -\text{cosec}^2 x$

2. Inverse Trigonometric Derivatives:

   • $\frac{d}{dx} \sin^{-1}x = \frac{1}{\sqrt{1-x^2}}$
   • $\frac{d}{dx} \cos^{-1}x = \frac{-1}{\sqrt{1-x^2}}$
   • $\frac{d}{dx} \tan^{-1}x = \frac{1}{1+x^2}$

3. Chain Rule:

   • $\frac{d}{dx}f(g(x)) = f'(g(x))g'(x)$, applied to functions like $\cos(x^2)$ and $\tan^2(x)$.

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Tips and Tricks

1. For first principle problems, focus on simplifying $\frac{\sin h}{h} \to 1$ and $\frac{\cos h - 1}{h} \to 0$ as $h \to 0$.
2. Memorize the standard derivatives for trigonometric and inverse trigonometric functions.
3. For composite functions like $\cos x^2$, apply the chain rule efficiently.
4. Be cautious about domain restrictions for inverse functions (e.g., $\sin^{-1}x$ is defined only for $|x| \leq 1$).

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Summary

This exercise enhances understanding of derivatives involving trigonometric and inverse trigonometric functions. It bridges foundational calculus principles with practical applications. Mastery of differentiation techniques from the first principle ensures conceptual clarity, while familiarity with standard formulas aids efficiency.

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Reference

By Sir Shahzad Sair:

By Great Science Academy:

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