Let $g \in C^1(\mathbb R)$ be a real valued function and $f$ defined by $$f(u,v,w) = \int_{u}^{v} g(w^2+\sqrt{s})\,\,ds$$ where $u,v,w \in \mathbb R$ and $u,v>0$.
Find all partial derivatives.
I'm not sure how to attempt this problem. I assume if it were a function of two variables, say something like $$f(x,w) = \int_{0}^{x} g(w^2+\sqrt{s})\,\,ds$$ then for example the partial derivative with respect to $x$ would just be $g(w^2 + \sqrt{x})$ (is that true?).
Anyhow, some hint or strategy would be very welcomed.
Answer
Call $h(s) = g(w^2-\sqrt{s})$. When you compute the partial derivatives with rispect to $u,v$ the variable $w$ is fixed, hence you can think $h$ as a function $\mathbb{R} \longrightarrow \mathbb{R}$.
So
$$\frac{\partial}{\partial u} \int_u^v h(s) ds = - \frac{\partial}{\partial u} \int_v^u h(s) ds =- h(u)$$
since $v$ is considered a constant. In the same way you get
$$ \frac{\partial}{\partial v} \int_u^v h(s) ds = h(v)$$
While for the third partial derivative you need to exchange the derivative with the integral sign, so you get
$$\frac{\partial}{\partial w} \int_u^v g(w^2-\sqrt{s}) ds =
\int_u^v \frac{\partial}{\partial w} g(w^2-\sqrt{s}) ds =
\int_u^v g'(w^2-\sqrt{s}) 2w \ ds$$
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