How to differentiate y = e x 2 ?

Question

How to differentiate       y    =          e                        x          2                    ?

Adriana Finley · Accepted Answer

Chain Rule - In order to differentiate a function of a function, say

y, = f (g (x))

, where we have to find

\frac{d y}{d x}

, we need to do (a) substitute

u = g (x)

, which gives us

y = f (u)

. Thus we need to use a formula called Chain Rule, which states that

\frac{d y}{d x} = \frac{d y}{d u} \times \frac{d u}{d x}

. In fact if we have something like

y = f (g (h (x)))

, we can have

\frac{d y}{d x} = \frac{d y}{d f} \times \frac{d f}{d g} \times \frac{d g}{d h}

Here we have

y = e^{u}

, where

u = x^{2}

Thus,

\frac{d y}{d x} = \frac{d y}{d u} \times \frac{d u}{d x}

=

\frac{d}{d u} e^{u} \times \frac{d}{d x} (x^{2})

=

e^{u} \times 2 x = 2 x e^{x^{2}}

How do you differentiate y=e^(x^2)?