Find dy/dx , if y = e^x^2+2 + sin (5x^2+2) + 3 cos^2x?

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Find dy/dx , if y = e^x^2+2 + sin (5x^2+2) + 3 cos^2x?

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Answer 1 · 2018-07-28T14:04:03

$\frac{d y}{d x} = 2 x e^{x^{2} + 2} + 10 x cos (5 x^{2} + 2) - 3 sin 2 x$ $dy/dx=2xe^{x^2+2}+10x\cos(5x^2+2)-3\sin2x$

Explanation:

Given function:

$y = e^{x^{2} + 2} + sin (5 x^{2} + 2) + 3 {cos}^{2} x$ $y=e^{x^2+2}+\sin(5x^2+2)+3\cos^2x$

differentiating above equation w.r.t. $x$ $x$ using chain rule as follows

$\frac{d y}{d x} = \frac{d}{d x} (e^{x^{2} + 2}) + \frac{d}{d x} sin (5 x^{2} + 2) + 3 \frac{d}{d x} {cos}^{2} x$ $dy/dx=d/dx(e^{x^2+2})+d/dx\sin(5x^2+2)+3d/dx\cos^2x$

$= e^{x^{2} + 2} \frac{d}{d x} (x^{2} + 2) + cos (5 x^{2} + 2) \frac{d}{d x} (5 x^{2} + 2) + 3 (2 cos x) \frac{d}{d x} (cos x)$ $=e^{x^2+2}d/dx(x^2+2)+\cos(5x^2+2)d/dx(5x^2+2)+3(2\cos x)d/dx(\cosx)$

$= e^{x^{2} + 2} (2 x) + cos (5 x^{2} + 2) (10 x) + 3 (2 cos x) (- sin x)$ $=e^{x^2+2}(2x)+\cos(5x^2+2)(10x)+3(2\cos x)(-\sinx)$

$= 2 x e^{x^{2} + 2} + 10 x cos (5 x^{2} + 2) - 3 sin 2 x$ $=2xe^{x^2+2}+10x\cos(5x^2+2)-3\sin2x$

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Find dy/dx , if y = e^x^2+2 + sin (5x^2+2) + 3 cos^2x?

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