Question

How do you use the chain rule to differentiate `y=root3(-2x^4+5)`?

Answer 1

`y'=(-8x^3)/(3root3((-2x+5)^2)` Or alternatively `y'=-(8x^3)/(3root3((5-2x)^2`

Explanation:

The first step is to rewrite the equation using powers:

`y=(-2x^4+5)^(1/3)`

Now we are able to apply the chain rule, we basically take the derivative of the outside times the derivative of the inside. You will need the power rule too.

`d/dx=1/3(-2x^4+5)^(-2/3)xxd/dx(-2x^4+5)`

`d/dx=color(blue)(1/3(-2x^4+5)^(-2/3))xx(-8x^3)`

What we want to do now is rewrite what's in blue:

`color(blue)(1/3(-2x^4+5)^(-2/3))=1/(3root3((-2x+5)^2)`

Now that we know this we can simply multiply straight through:

`d/dx=1/(3root3((-2x+5)^2))xx(-8x^3)/1`

Our final answer is:

`y'=(-8x^3)/(3root3((-2x+5)^2)` Or `y'=-(8x^3)/(3root3((5-2x)^2`