# Partial Derivative of a Definite Integral

1. Feb 25, 2016

### zl99

I'm trying to find the partial derivatives of:
f(x,y) = ∫ (from -4 to x^3y^2) of cos(cos(t))dt

and I am completely lost, any help would be appreciated, thanks.

2. Feb 25, 2016

### Staff: Mentor

Calculate the integral, i.e. the anti-derivative of $cos^2(t)$, substitute the boundaries and differentiate it.

3. Feb 25, 2016

### Staff: Mentor

The integrand that zl99 wrote isn't $\cos^2(t)$ -- it's $\cos(\cos(t))$.

I'd say that the strategy here is to use a form of the Fundamental Theorem of Calculus; i.e., that $\frac d {dt} \int_a^x f(t)~dt = f(x)$. In this problem, I think you need to involve the chain rule. I haven't worked the problem, but that's the way I would go.

4. Feb 26, 2016

### Ssnow

yes as @Mark44 said you must before use the FTC (Fundamental Theorem of Calculus) and multiply by the partial derivatives of $x^3y^2$ in one case you obtain the partial derivative respect $x$ and in the other case respect $y$ (you will use the chain rule for this)