Chain Rule for Derivatives. Example Problem
d/dx((3x^2 -5)^4)=
We can take the derivative of 3x squared minus 5 all raised to the fourth power by applying the chain rule. The clue here that we're going to use the chain rule is the fact that this is not a nice polynomial function, it's not the product of two functions, it's not the quotient of two functions. Instead, what we have is we have something that is in parentheses is being raised to an exponent. Basically, we have a composite function. This looks like u to the fourth power. So that means this is a part u, which is 3x squared minus 5." The derivative of that is 6x.
When you have a comparative function, it is often easier to use the chain rule. In this example, you can use the chain rule to find the derivative of u to the fourth power. The derivative of u is equal to four times x cubed, minus five cubed times × squared. There is no need to multiply out this expression because we already know how to simplify exponents with the distributive property.
d/dx (3x^2 -5)^4
u=3x^2 -5
du=6x
4(3x^2 -5)^3 *6x=24x(3x^2 -5)^3
