Question

How do you find the first and second derivative of `y=e^(x^3)`?

Answer 1

`d/(dx) e^(x^3)= 3x^2e^(x^3)`
`d^((2))/(dx^2) e^(x^3)=6xe^(x^3)(1+3/2x^3)`

Explanation:

Using the chain rule:

`d/(dx) e^(x^3) = (de^(x^3))/(d(x^3)) * (d(x^3)) /(dx) = 3x^2e^(x^3)`

Then using the product rule:

`d^((2))/(dx^2) e^(x^3) =d/(dx)(3x^2e^(x^3)) = (d/(dx)3x^2)e^(x^3)+ 3x^2(d/(dx) e^(x^3)) = 6xe^(x^3)+3x^2*3x^2e^(x^3)=6xe^(x^3)+9x^4e^(x^3)=6xe^(x^3)(1+3/2x^3)`