How do you use the chain rule to differentiate y=root3(-2x^4+5)y=32x4+5?

1 Answer
Jun 12, 2017

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