I'm studying for a test, and I don't understand how to do this problem. Could anyone help me?

Find the derivative dy/dx if y equals the integral from sin(x) to 5 of 1/u^3 du. Be sure to show your work and give the statement of any theorems that you use.

2. I can tell you the answer: $\displaystyle -\cot x \csc^2 x$.

3. Originally Posted by juicysharpie
I'm studying for a test, and I don't understand how to do this problem. Could anyone help me?

Find the derivative dy/dx if y equals the integral from sin(x) to 5 of 1/u^3 du. Be sure to show your work and give the statement of any theorems that you use.
$\displaystyle y = \int_{\sin x}^5 \frac{1}{u^3} \, du = - \int^{\sin x}_5 \frac{1}{u^3} \, du$.

Let $\displaystyle t = \sin x$ and use the chain rule:

$\displaystyle \frac{dy}{dx} = \frac{d}{dt} \left[- \int^{t}_5 \frac{1}{u^3} \, du \right] \cdot \frac{dt}{dx}$

$\displaystyle = -\frac{1}{t^3} \cdot \cos x = -\frac{1}{\sin^3 x} \cdot \cos x = \frac{\cos x}{\sin^3 x}$.

Your job is to identify and name the theorems that I used.

4. Originally Posted by mr fantastic
$\displaystyle = -\frac{1}{t^3} \cdot \cos x = -\frac{1}{\sin^3 x} \cdot \cos x = \frac{\cos x}{\sin^3 x}$.
Looks good except you dropped a minus sign on the last step.

5. Originally Posted by mr fantastic
$\displaystyle \frac{dy}{dx} = \frac{d}{dt} \left[- \int^{t}_5 \frac{1}{u^3} \, du \right] \cdot \frac{dt}{dx}$
I'm sorry, could you please explain how you got to this step? I don't quite understand how you did it.

Thanks for helping!!

6. Originally Posted by juicysharpie
I'm sorry, could you please explain how you got to this step? I don't quite understand how you did it.

Thanks for helping!!
Review chain rule in your class notes or textbook.