Finding partial derivative of an integral, no need to use chain rule?

Question

I need help for this question:

Find the partial derivatives of this function

$$ f(x,y)= \int_x^y \cos(−7t^2+4t−5)dt $$

This was what I did:

$$ f(x,y)= \int_0^x \cos(−7t^2+4t−5) \text{dt} - \int_0^y \cos(−7t^2+4t−5)\text{dt} $$

So $f_x(x,y)$ would be $\cos(-7x^2+4x-5) \frac{d}{dx} \cos(-7x^2+4x-5)$

And likewise for $f_y(x,y)$

But the answer provided by my teacher is just $f_x(x,y)=\cos(-7x^2+4x-5)$, without using the chain rule. Why is that?

Answer 1

Firstly, when you wrote $f$ as a difference of two integrals, you got them the wrong way round. (However, given what your teacher claimed is the correct answer, my guess is you actually transposed the limits of the original integral defining $f$.) Secondly, the Fundamental Theorem of Calculus gives $\frac{d}{dx}\int_0^xg(t)dt$ as $g(x)$, not $g(x)g^\prime(x)$, which would be $\frac{d}{dx}\left(\frac12 g^2\right)$.

Answer 2

Hint: Use just Leibniz integral rule