Question

Question #a62bb

Answer 1

`dy/dx=15x^2(1+x^3)^4`

Explanation:

.

`y=(1+x^3)^5`

`u=1+x^3`

`(du)/dx=3x^2`

`y=u^5`

`dy/(du)=5u^4`

Using Chain Rule:

`dy/dx=dy/(du)*(du)/dx`

`dy/dx=5u^4(3x^2)`

`dy/dx=5(1+x^3)^4(3x^2)`

`dy/dx=15x^2(1+x^3)^4`

Answer 2

`f'(x)=15x^14+60x^11+90x^8+60x^5+15x^2`

Explanation:

You need to know two rules: the power rule and the chain rule.
The power rule states that if `f(x)=x^n`, then `f'(x)=nx^(n-1)`
The chain rule states that if `f(x)=g(x)` , then the derivative of `f(x)` is the derivative of `g(x)` times the derivative of `x`.

Therefore, if `f(x)=(1+x^3)^5`, then
`f'(x)=5(1+x^3)^4xx(3x^2)` or
`f'(x)=15x^2(1+x^3)^4`
You could of course expand this to get:
`f'(x)=15x^14+60x^11+90x^8+60x^5+15x^2`

We're of course, saying that `f(x)=y`