# How do you find the derivative of y=e^(x^2) ?

Sep 7, 2014

You could do this problem by using the chain rule. You have a function "within a function", so by the chain rule, you would derive the "outer" function first (while leaving the inner function alone) and then you would times that by the derivative of the "inner" function.

Mathematically,
$F ' \left(x\right) = f ' \left(g \left(x\right)\right) \left(g ' \left(x\right)\right)$

So in this case ${e}^{{x}^{2}}$ would the the "outer" function. ${x}^{2}$ would be the "inner" function that is composed in the ${e}^{{x}^{2}}$

Let's derive the outer function first (while leaving the inner function alone, that is):

$\frac{d}{\mathrm{dx}} {e}^{{x}^{2}} = {e}^{{x}^{2}}$ (note that this statement isn't actually true- it's just showing the step of deriving the outer function while leaving the inner function alone. The actual derivative is after you derive the inner function as well and multiply by it)

Deriving the inner function:

$\frac{d}{\mathrm{dx}} {x}^{2} = 2 x$

Combining the two through multiplication, the actual derivative is:

$\frac{d}{\mathrm{dx}} {e}^{{x}^{2}} = 2 x {e}^{{x}^{2}}$