Question

Differentiating Exponential functions In Exercise,find the derivative of the function. See Example 2 and 3.

Answer 1

Answer: $-2xe^{-x^2}$

Step-by-step explanation:

Let u be a differentiable function , then

$\dfrac{d}{dx}(e^x)=e^x$

$\dfrac{d}{dx}e^u=e^u\dfrac{du}{dx}$              (1)

Given function : $y = e^{-x^2}$

Differentiate both sides with respect to x , we get

$y'=e^{-x^2}\dfrac{d(-x^2)}{dx}$  (By using  (1))

$\Rightarrow\ y'=e^{-x^2}(-2x)$  [∵ $\dfrac{d}{dx}(x^n)=x^{n-1}$]

$\Rightarrow\ y'=-2xe^{-x^2}$

Hence, the derivative of the given function is $-2xe^{-x^2}$ .