What is the derivative of sin x^2?

Apr 22, 2018

$2 x \cos \left({x}^{2}\right)$

Explanation:

Using the chain rule,

$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{\mathrm{dy}}{\mathrm{du}} \cdot \frac{\mathrm{du}}{\mathrm{dx}}$

Let $u = {x}^{2} , \mathrm{du} = 2 x \setminus \mathrm{dx} , \therefore \frac{\mathrm{du}}{\mathrm{dx}} = 2 x$.

Then $y = \sin \left(u\right) , \mathrm{dy} = \cos \left(u\right) \setminus \mathrm{du} , \therefore \frac{\mathrm{dy}}{\mathrm{du}} = \cos \left(u\right)$.

Combining, we get,

$\frac{\mathrm{dy}}{\mathrm{dx}} = \cos \left(u\right) \cdot 2 x$

$= 2 x \cos \left(u\right)$

Substitute back $u = {x}^{2}$ to get the final answer:

$= 2 x \cos \left({x}^{2}\right)$