Find the second-order derivative of the funct | vedclass

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Question

(A) Let $y = e^{x} \sin 5x$.Applying the product rule for differentiation:$\frac{dy}{dx} = \frac{d}{dx}(e^{x}) \cdot \sin 5x + e^{x} \cdot \frac{d}{dx

Vedclass · Accepted Answer

$2 e^{x}(5 \cos 5x - 12 \sin 5x)$

Vedclass · Answer

(A) Let $y = e^{x} \sin 5x$.Applying the product rule for differentiation:$\frac{dy}{dx} = \frac{d}{dx}(e^{x}) \cdot \sin 5x + e^{x} \cdot \frac{d}{dx}(\sin 5x)$$= e^{x} \sin 5x + e^{x} \cdot (5 \cos 5x) = e^{x}(\sin 5x + 5 \cos 5x)$.Now, differentiate again with respect to $x$:$\frac{d^2y}{dx^2} = \frac{d}{dx}[e^{x}(\sin 5x + 5 \cos 5x)]$$= \frac{d}{dx}(e^{x}) \cdot (\sin 5x + 5 \cos 5x) + e^{x} \cdot \frac{d}{dx}(\sin 5x + 5 \cos 5x)$$= e^{x}(\sin 5x + 5 \cos 5x) + e^{x}(5 \cos 5x - 25 \sin 5x)$$= e^{x}(\sin 5x + 5 \cos 5x + 5 \cos 5x - 25 \sin 5x)$$= e^{x}(10 \cos 5x - 24 \sin 5x)$$= 2e^{x}(5 \cos 5x - 12 \sin 5x)$.

Find the second-order derivative of the function $y = e^{x} \sin 5x$.

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