Question

Differentiate the following w.r.t.x. :

y = (x² + 2)² sin x

Answer 1

y = (x² + 2)² sin x

Differentiating w.r.t. x, we get

${\displaystyle \frac{\text{d}y}{\text{d}x}=\frac{\text{d}}{\text{d}x}\left[{(x^2+2)}^2\sin x\right]}$

${\displaystyle \frac{\text{d}y}{\text{d}x}={(x^2+2)}^2\frac{\text{d}}{\text{d}x}\sin x+\sin x\left[\frac{\text{d}}{\text{d}x}{(x^2+2)}^2\right]}$

= ${\displaystyle {(x^2+2)}^2\cos x+\sin x\frac{\text{d}}{\text{d}x}(x^4+4x^2+4)}$

= ${\displaystyle {(x^2+2)}^2\cos x+\sin x(4x^3+8x)}$

= ${\displaystyle {(x^2+2)}^2\cos x+\sin x\cdot \left(4x\right)(x^2+2)}$

= ${\displaystyle (x^2+2)[(x^2+2)\cos x+4x\sin x]}$