Question

$\frac{d}{dx} \sin(x {}^{2} + 1)$
Please solve this problem. Irrelevant answers will be reported.​

Answer 1

$\displaystyle\huge\red{\underline{\underline{Solution}}}$

Let

$y= sin( {x}^{2} + 1) \: \: .........(1)$

Also let

$z \: = {x}^{2} + 1 \: \: .......(2)$

From Equation (1) using Equation (2)

$y = sinz \: \: \: \: \: ........(3)$

Differentiating both sides of Equation (2) with respect to x we get

$\displaystyle \: \frac{dz}{dx} = 2x$

Differentiating both sides of Equation (3) with respect to z we get

$\displaystyle \: \frac{dy}{dz} = cosz$

Hence

$\displaystyle \: \frac{dy}{dx} = \frac{dy}{dz} \times \frac{dz}{dx}$

$\therefore \: \displaystyle \: \frac{dy}{dx} = 2x \: cosz = 2x \: cos( {x}^{2} + 1)$