# How do you use the chain rule to differentiate #y=sqrt(1/(2x^3+5))#?

##### 2 Answers

#### Explanation:

When differentiating functions with the chain rule, it helps to think of our function as "layered," remembering that we must differentiate one layer at a time, from the outermost layer to the innermost layer, and multiply these results.

Here, our outer layer would be the square root, while the inner layer would be the quotient of a polynomial.

Let's temporarily denote everything inside the root by

Let's rewrite this in terms of

We've differentiated our outer layer. Moving on to the inner layer,

Let's multiply our differentiated layers together:

Simplify:

Using the division rule for exponents, we get:

See below

#### Explanation:

So, for square roots and other nth-root functions, I personally always convert them to rational exponents. There are probably other ways to do it, but the students I help always seem to like this method, too.

So the function

I'll even go one further by changing the inside to a negative exponent so I can avoid using a quotient rule:

This leads us to something interesting. We can actually multiply the exponents (taking a power to a power), so now we have:

So here's where we use the chain rule for differentiation. Basically, we take the derivative of the outside-most function multiplied by the derivative of the inside function. I like to do this in steps so I don't get confused.

Simplifying and converting negative exponents to positive exponents gives us: