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: