Find the derivative of the function. y = [x + (x + sin^2(x))^5]^4

Answer 1

So this is a crazy chain rule problem.

Take the derivative of the outside and multiply it by the derivative of the inside, repeat this multiplication until you just take the derivative of x.

First derivative is (____) ^4

This becomes 4 (____) ^3

The blank is x + (x + sin^2(x))^5

Now we take the derivative of the blank which is

1 + 5 (x + sin^2(x))^4

Now the derivative of the inside again.

x + sin^2(x) becomes

1 + 2 sin(x) * d/dx (sin(x)

Notice that the function sin(x) is inside the function sin^2(x)

d/dx sin(x) is cos(x)

That was a lot so now we multiply all the derivative answers together.

I'll rewrite them here.

4(x + (x + sin^2(x))^5)^3

1 + (5 (x + sin^2(x))^4) (1 + 2 sin(x) cos(x))

Put parenthesis around both of these terms so that they are being multiplied together.

NOTICE that we only multiply together the terms that go inside the others.

1 + 2sin(x) cos(x)

Instead of

(1+ 2sin(x)) cos(x)

This is because the 1 is not within the previous function.

This is the solution.

It might be a bit messy to tell what is going on. Message me if you have any questions.