Question

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

Answer 1

`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.