Question

How do you differentiate `f(x)=e^x*sin(5x^2)` using the product rule?

Answer 1

`e^x(sin(5x^2)+10xcos(5x^2))`

Explanation:

Let `u=e^x` and `v=sin(5x^2)`
Therefore `du/dx=e^x`
and to find `dv/dx` you need to use chain rule...
so... let `w=5x^2` then `v=sin(w)`
`dv/dw = cos(w)`
and `dw/dx=10x`
so the differential of `v=sin(5x^2)` is `10xcos(5x^2)`

from there you use the rest of product rule so... `dy/dx=(v*du/dx)+(u*dv/dx)`
so...
`(sin(5x^2)*e^x)+(e^x*10xcos(5x^2))`
`e^x(sin(5x^2)+10xcos(5x^2))`

Answer 2

`f'(x)=e^x(10xcos(5x^2)+sin(5x^2))`

Explanation:

`"Given "f(x)=g(x)h(x)" then"`

`f'(x)=g(x)h'(x)+h(x)g'(x)larrcolor(blue)"product rule"`

`g(x)=e^xrArrg'(x)=e^x`

`h(x)=sin(5x^2)larrcolor(blue)"use chain rule"`

`rArrh'(x)=cos(5x^2)xxd/dx(5x^2)`

`color(white)(rArrh'(x))=10xcos(5x^2)`

`rArrf'(x)=10xcos(5x^2)e^x+sin(5x^2)e^x`

`color(white)(rArrf'(x))=e^x(10xcos(5x^2)+sin(5x^2))`