# How to calculate the derivative of following integral?

I did an exercise which would like to calculate the derivative in $$x$$ of the integral $$T(x) = \int\limits_0^{1} t^2 (1-t)^3 (t-x)^4 dt,$$ Of course, I showed that $$\dfrac{d}{dx} T(x) = - 4 \int\limits_0^{1} t^2 (1-t)^3 (t-x)^3 dt .$$

The problem is that I'd like a general resultat which can give a method to determinate the derivative in $$x$$ of the integral $$T_1(x) = \int\limits^1_0 f(t, x) dt$$ where $$f(t,x)$$ is a polynomial function of $$t$$ and $$x$$.

My question: Is there a resultat to determinate the derivative in $$x$$ of the integral $$T_1(x) = \int\limits^1_0 f(t, x) dt$$ where $$f(t,x)$$ is a polynomial function of $$t$$ and $$x$$?

Thank you very much for your interests!

• Have you learned about partial derivatives? You can do this by Leibniz's rule for differentiation under the integral sign. – saulspatz Apr 4 at 2:25
• Multiply out the polynomials. You should get the same result regardless of whether you differentiate with respect to $x$ before or after you integrate. – Doug M Apr 4 at 2:38
• @saulspatz Yes, it is nice! Thank you so much! – mathJuan Apr 4 at 2:39

## 1 Answer

In general, if $$a$$ and $$b$$ are constants, and $$F(x)=\int_a^b f(x,t)dt$$ then $$F'(x)=\int_a^b\frac{\partial}{\partial x}f(x,t)dt$$ for a sufficient function $$f$$. Here $$\frac{\partial}{\partial x}$$ denotes the partial derivative. More details can be found here.