I have a function I would like to differentiate but am wondering if I my method is allowable:
If $\displaystyle f(x)= \int_{a}^{b} h(t) \:\mathrm{d}t$, what is the derivative of $f$ with respect to $x$, if $x$ occurs in the expression of $h(t)$.
Can I solve this by simply taking the derivative of $h(t)$ as I would normally any function while treating instances of $t$ as constants? Or must I account for the integration before taking the derivative. I understand that if I could compute the integral I would end up with an expression for $f(x)$ in terms of $x$ and that the integral is kind of just a place holder for that expression, but I am unsure of whether or not taking the derivative before computing the integral would change the result. Any advice would be appreciated as well as any suggested readings on this type of problem!
Best Answer
Yes, you can, assuming some weak conditions are met. If $h(x,t)$ has continuous partial derivatives, then $$\frac{\mathrm{d}}{\mathrm{d}x} \left (\int_{a}^{b}h(x,t)\,\mathrm{d}t \right) = \int_{a}^{b} \frac{\partial}{\partial x}h(x,t)\; \mathrm{d}t.$$