Question

How do you differentiate `y=e^(2x^3)`?

Answer 1

`(dy)/(dx)=6x^2e^(2x^3)`

Explanation:

We use Chain Rule here.

In order to differentiate a function of a function, say `y, =f(g(x))`, where we have to find `(dy)/(dx)`, we need to do substitute `u=g(x)`, which gives us `y=f(u)`.

The Chain Rule states that `(dy)/(dx)=(dy)/(du)xx(du)/(dx)`.

In fact if we have something like `y=f(g(h(x)))`, we can have `(dy)/(dx)=(dy)/(df)xx(df)/(dg)xx(dg)/(dh)`

Here we have `y=e^(2x^3)`

Hence `(dy)/(dx)=(d(e^(2x^3)))/(d(2x^3))xx(d(2x^3))/(dx)`

= `e^(2x^3)xx2xx3x^2`

= `6x^2e^(2x^3)`