Question

How do you differentiate `f(x)=e^(5x^2+x+3)` using the chain rule?

Answer 1

`f'(x) = (10x + 1 )e^(5x^2+x+3)`

Explanation:

Using the `color(blue)" chain rule "`

`d/dx[f(g(x)) ] = f'(g(x)).g'(x)`

and `d/dx(e^x) = e^x`

f'(x)`= e^(5x^2+x+3) d/dx(5x^2 + x + 3)`

`= e^(5x^2+x+3)(10x + 1 )`

Answer 2

`h'(x)=e^(5x^2+x+3)(10x+1)`

Explanation:

The given equation is `h(x)=e^(5x^2+x+3)`. But it seems as if it is a function inside a function. So, let's write `g(x)=5x^2+x+3` and `f(x)=e^x`.
So that means `f(g(x))=e^(g(x))=e^(5x^2+x+3`

From chain rule, we have `d/dx(f(g(x))=h'(x)=f'(g(x))* g'(x)`.
So, taking for `f(g(x))` and `g(x)`, we get
`h'(x)=e^(g(x))*d/dx(g(x))==e^g(x)(10x+1)`

You can substitute it all to get back the proper answer.