# What are the critical values, if any, of #f(x) = f(x) = x^{2}e^{15 x}#?

##### 1 Answer

Nov 4, 2015

#### Explanation:

To find the critical points, we need the first derivative. This function is a **multiplication** of a **power** and a **composite exponential**. Let's see how to deal with these three things:

- The derivative of a multiplication
#f*g# is (Leibniz formula)#f'*g + f*g'# ; - The derivative of a power
#x^n# is#nx^{n-1}# ; - The derivative of a composite function
#f(g(x))# is#f'(g(x)) * g'(x)# ; - The derivative of an exponential
#e^x# is the exponential itself.

Let's put all these things together:

- The derivative of
#x^2 e^{15x}# is

- The derivative of
#x^2 is # 2x^1=2x#, and the expression becomes

- The exponential is a composite function, so we must derive the exponential and then multiply for the derivative of the exponent:

So, the answer is

We can factor an exponential and a