Question

Finding the derivative using the fundamental theorem of calculus?

Answer 1

The fundamental theorem says that
\[\frac{d}{dx}\left[\int_{c}^{g(x)}f(t)~dt\right]=f(g(x))\cdot g'(x)\]
where \(c\) is a constant.

Try writing your integral in this way, then take the derivative.

Answer 2

Naavidya: To help me (and others) help you: would you mind typing in one example of use of the Fundamental Theorem in finding the derivative of a function defined as an integral?

The example you've illustrated is a bit more challenging than earlier problems of this nature, because both limits of integration are in themselves functions of x.

Answer 3

Right, and
\[\int_a^b=-\int_b^a\]

Answer 4

Exactly! Great start!
And beyond that, you can rewrite the 2nd integral with limits of integration in reverse order, also changing the sign in front of it to (+) from (-). Mind typing that to ensure that we're on the same wavelength?

Answer 5

And then the furthest I got was to
\[\cos(5x)^7*(5)-\cos(cosx)^7*(-sinx)\]but I don't really know how to simply further

Answer 6

I see no need to simplify further. You've done a great job.
You could, if you wish, write that 5 in front of the first cos term, and change the sign of the 2nd term to (+) because (-)(-)=(+).

Answer 7

It's not accepting the answer for some reason so I guess I entered it wrong or it wants it in some other format...

Answer 8

I see nothing wrong mathematically with your response. You could try moving the (-sin x) factor in front of cos(cos x)^7, but this does not change anything mathematically.

Answer 9

I can't see anything wrong with it either...

Answer 10

I'd suggest checking with a classmate, or even with your teacher if possible, but moving on right now. But of course it's up to you to decide what to do next.
You could also put the problem aside for a while and retry it later, from scratch.