2.7 Q-4

Question Statement

Find $y^{4}$ for the following functions:

• i. $y = \sin(3x)$
• ii. $y = \cos^3(x)$
• iii. $y = \ln(x^2 - 9)$

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Background and Explanation

To solve this problem, you need to use basic calculus concepts like differentiation. The goal is to differentiate the given functions four times to find $y^4$. We’ll use the chain rule, product rule, and other differentiation techniques where necessary.

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Solution

i. $y = \sin(3x)$

To differentiate $y = \sin(3x)$, we apply the chain rule.

• First derivative:

$$y_1 = \frac{d}{dx} [\sin(3x)] = 3 \cos(3x)$$

• Second derivative:

$$y_2 = \frac{d}{dx} [3 \cos(3x)] = -9 \sin(3x)$$

• Third derivative:

$$y_3 = \frac{d}{dx} [-9 \sin(3x)] = -27 \cos(3x)$$

• Fourth derivative:

$$y_4 = \frac{d}{dx} [-27 \cos(3x)] = 81 \sin(3x)$$

Answer: $y_4 = 81 \sin(3x)$

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ii. $y = \cos^3(x)$

For $y = \cos^3(x)$, we’ll use the chain rule and product rule to differentiate.

• First derivative:

$$y_1 = \frac{d}{dx} [\cos^3(x)] = -3 \sin(x) \cos^2(x)$$

• Second derivative:

$$y_2 = \frac{d}{dx} [-3 \sin(x) \cos^2(x)] = -3 \cos(x) + 9 \sin^2(x) \cos(x)$$

• Third derivative:

$$y_3 = \frac{d}{dx} [-3 \cos(x) + 9 \sin^2(x) \cos(x)] = -6 \sin(x) \cos^2(x) - 27 \sin(x) \cos(x)$$

• Fourth derivative:

$$y_4 = \frac{d}{dx} [-6 \sin(x) \cos^2(x) - 27 \sin(x) \cos(x)] = -60 \cos(x) + 81 \cos^3(x)$$

Answer: $y_4 = -60 \cos(x) + 81 \cos^3(x)$

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iii. $y = \ln(x^2 - 9)$

For $y = \ln(x^2 - 9)$, we use the chain rule and differentiate multiple times.

• First derivative:

$$\frac{dy}{dx} = \frac{d}{dx} \left[ \ln(x^2 - 9) \right] = \frac{1}{x^2 - 9} \cdot 2x = \frac{2x}{x^2 - 9}$$

• Second derivative:

$$y_2 = \frac{d}{dx} \left[ \frac{2x}{x^2 - 9} \right] = -\frac{2(x^2 + 9)}{(x^2 - 9)^2}$$

• Third derivative:

$$y_3 = \frac{d}{dx} \left[ -\frac{2(x^2 + 9)}{(x^2 - 9)^2} \right] = \frac{12x(x^2 + 9)}{(x^2 - 9)^3}$$

• Fourth derivative:

$$y_4 = \frac{d}{dx} \left[ \frac{12x(x^2 + 9)}{(x^2 - 9)^3} \right] = -6 \left[ \frac{1}{(x + 3)^4} + \frac{1}{(x - 3)^4} \right]$$

Answer: $y_4 = -6 \left[\frac{1}{(x + 3)^4} + \frac{1}{(x - 3)^4}\right]$

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Key Formulas or Methods Used

• Chain Rule:

$$\frac{d}{dx} [f(g(x))] = f'(g(x)) \cdot g'(x)$$

• Product Rule:

$$\frac{d}{dx} [u(x)v(x)] = u'(x)v(x) + u(x)v'(x)$$

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Summary of Steps

1. Differentiate the given function using the chain rule and product rule.
2. Compute the first, second, third, and fourth derivatives.
3. Simplify each derivative step-by-step.
4. Write the final expression for $y_4$.