How do you differentiate #f(x) = sin((− x^2 − 1)^2) *(x^2 − 9)^2# using the chain rule?
To differentiate your function, you want to start by looking, overall, at what type of function you have. In short, you have a product (with chains inside). This could get a bit messy, haha!
Recall that chain rule looks like this:
First, start off with the product rule where if:
Note that the order of multiplication doesn't matter!
Starting off with your derivative:
Differentiate the first part:
The derivative of sin(x) is cos(x), but your "x" is much more complicated; it's a chain of functions and thus, by Chain Rule, you must multiply by the inside of the function. Therefore the derivative of the first part is:
Note that you had to do chain rule on the
So far we have:
Now we need to differentiate the second half of the equation. This is a simple chain rule problem.
By Chain Rule / Power Rule the derivative is:
Now you just multiply by the first part of the function unchanged to get:
Finally, combine everything to get:
I know this was a bit messy, but hopefully everything was clear! Chain rule, combined with the other rules for derivatives can get quite complicated, it just requires practice! Hopefully this helps! :)