07_Ex 2.7

Exercise Questions

Questions	Question Links .
Q1. Find $y^{2}$ if…	2.7 Q-1
Q2. Find $y_{2}$ if…	2.7 Q-2
Q3. Find $y_{2}$ if…	2.7 Q-3
Q4. Find $y^{4}$ if	2.7 Q-4
Q5. if $x=\operatorname{Sin} \theta, y=\operatorname{sinm} \theta$ show that…	2.7 Q-5
Q6. If $y=e^{x} \operatorname{Sin} x$ show that $\frac{d^{2} y}{d x^{2}}=\frac{2 d y}{d x}+2 y=0$	2.7 Q-6
Q7. If $y=e^{a x} \operatorname{Sin} b x$ show that $\frac{d^{2} y}{d x^{2}}-2 a \frac{d y}{d x}+\left(a^{2}+b^{2}\right) y=0$	2.7 Q-7
Q8. If $y=\left(\operatorname{Cos}^{-1} x\right)^{2}$ prove that $\left(1-x^{2}\right) y_{2}-x y_{1}-2=0$	2.7 Q-8
Q9. if $y=a \operatorname{Cos}(\ln x)+b \operatorname{Sin}(\ln x)$	2.7 Q-9

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Overview

This exercise focuses on the concept of higher-order derivatives. Higher-order derivatives are the successive derivatives of a function, such as the second, third, or nth derivatives. These derivatives provide insight into the rate of change of the function and its curvature, which are important in physics, engineering, and economics. In this exercise, you’ll apply rules of differentiation (such as the Power Rule, Product Rule, Quotient Rule, and Chain Rule) to differentiate various functions multiple times.

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Key Concepts

• Higher-order Derivatives: These are the derivatives of a function after the first derivative. For example, the second derivative is the derivative of the first derivative, and so on. Higher-order derivatives describe the rate of change of the rate of change, which is important for understanding acceleration, concavity, and other phenomena.

• Power Rule: This rule is used for differentiating expressions involving powers of $x$. For any constant $n$, the derivative of $x^n$ is $nx^{n-1}$.

• Product Rule: When differentiating a product of two functions, the derivative is given by $\frac{d}{dx}(f(x)g(x)) = f'(x)g(x) + f(x)g'(x)$.

• Quotient Rule: When differentiating a quotient of two functions, the derivative is given by $\frac{d}{dx}\left(\frac{f(x)}{g(x)}\right) = \frac{f'(x)g(x) - f(x)g'(x)}{g(x)^2}$.

• Chain Rule: Used for differentiating composite functions, i.e., functions within functions. The derivative is given by $\frac{d}{dx}f(g(x)) = f'(g(x))g'(x)$.

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Important Formulas

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

• Product Rule:
  $\frac{d}{dx}(f(x)g(x)) = f'(x)g(x) + f(x)g'(x)$

• Quotient Rule:
  $\frac{d}{dx}\left(\frac{f(x)}{g(x)}\right) = \frac{f'(x)g(x) - f(x)g'(x)}{g(x)^2}$

• Chain Rule:
  $\frac{d}{dx}f(g(x)) = f'(g(x))g'(x)$

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Tips and Tricks

• Simplify Early: Simplifying a function before differentiating can make the differentiation process much easier. For example, expand any polynomials and simplify fractions before applying the differentiation rules.
• Keep Track of Powers: Pay attention to the powers of $x$ when applying the Power Rule. This is essential for correctly calculating the derivatives.
• Apply the Rules Step-by-Step: If a function involves a combination of multiple differentiation rules (e.g., product, quotient, or chain rule), break it down step-by-step and apply each rule carefully.
• Use the Chain Rule for Nested Functions: Whenever you encounter a function within another function (e.g., $(2x + 5)^{3/2}$), apply the Chain Rule to differentiate the outer function first, then multiply by the derivative of the inner function.

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Summary

In this exercise, we applied the rules of differentiation (Power Rule, Product Rule, Quotient Rule, and Chain Rule) to find higher-order derivatives of various functions. We differentiated functions multiple times, each time simplifying the result, to find the second, third, and higher derivatives. These derivatives provide valuable information about the rate of change and concavity of the original function.

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Reference

By Sir Shazad Sair:

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