Question

How do you differentiate `y=e^(x^5+3)`?

Answer 1

The derivative of `e^x` is itself, `e^x`. So, when we have a function embedded inside `e^x`, like how here `x` is replaced by `x^5+3`, the derivative of `e^f(x)` where `f` is any other function is given by `e^f(x)*f'(x)`. We multiply by the derivative of `f` due to the chain rule.

In other words:

`d/dxe^x=e^x`

`d/dxe^u=e^u(du)/dx`

Regardless, we see that

`y=e^(x^5+3)`

`dy/dx=e^(x^5+3)d/dx(x^5+3)`

`=5x^4e^(x^5+3)`