# How do you find the derivatives of the function f(x) = sin(x^2)?

Apr 30, 2015

Use the chain rule, which says, for the sine function:

$\frac{d}{\mathrm{dx}} \left(\sin u\right) = \cos u \cdot \frac{\mathrm{du}}{\mathrm{dx}}$

(Or, if you prefer function notation:

For $f \left(x\right) = \sin \left(g \left(x\right)\right)$, the derivative is:

$f ' \left(x\right) = \cos \left(g \left(x\right)\right) \cdot g ' \left(x\right)$

So for this $f \left(x\right) = \sin \left({x}^{2}\right)$, we get

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

The question asks about the derivatives (plural). I'm not sure what you mean.
Do you also need the second derivative?, or a formula for the ${n}^{t h}$ derivative?, or perhaps it was just an error typing the question.