Question

How do I find `f'(x)` for `f(x)=x^2*10^(2x)` ?

Answer 1

The derivative of `f(x)=x^2cdot10^{2x}` is

`f'(x)=2x(1+xln10)10^{2x}`

Let us look at some details.

We need the following tools in your toolbox.

1. Power Rule: `(x^n)'=nx^{n-1}`

2. Exponential Rule: `(b^x)'=(lnb)b^x`

3. Product Rule: `[f(x)cdot g(x)]'=f'(x)cdot g(x)+f(x)cdot g'(x)`

4. Chain Rule: `[f(g(x))]'=f'(g(x))cdot g'(x)`

Let us find `(10^{2x})'` first.

By Chain Rule and Exponential Rule,

`(10^{2x})'=(ln10)10^{2x}cdot(2x)'=2(ln10)10^{2x}`

Now, we can find `f'(x)`.

By Product Rule,

`f'(x)=(x^2)'cdot10^{2x}+x^2cdot(10^{2x})'`

by Power Rule and the derivative we found above,

`=2xcdot 10^{2x}+x^2cdot2(ln10)10^{2x}`

by factoring out `2x10^{2x}`,

`=2x(1+xln10)10^{2x}`