How do you use the chain rule to differentiate y=root4(-3x^4-2)?

1 Answer
Jul 6, 2017

d/dxroot(4)(-3x^4-2)=(-3x^3)/root(4)((-3x^4-2)^3)

Explanation:

First recognize that the chain-rule means if you have a function within a function, you derive the "outer" function and then the "inner" function. To put this in a mathematical form:

d/dxf(g(x))=f'(g(x))*g'(x)

So first, let's look at root(4)(-3x^4-2). Notice that we can write this in its exponent form to get a clear image: (-3x^4-2)^(1/4).

First we derive the outside function or the exponent in this case. So we get 1/(4(-3x^4-2)^(3/4)).

Then we have to recognize that there's a function within the root: (-3x^4-2). Deriving this, we get (-12x^3).

Then using the chain-rule as a guide, we will multiply g'(x) or (-12x^3) in this case with f'(x):

1/(4(-3x^4-2)^(3/4))*(-12x^3)

Notice we can simplify the constant -12/4 to -3.

So our final answer is: (-3x^3)/root(4)((-3x^4-2)^3)

Note: you can write: root(4)((-3x^4-2)^3) as (-3x^4-2)^(3/4) for the final answer in the denominator.