Originally Posted by

**HallsofIvy** T is a function of the two variables, x and t, so I presume you mean the partial derivative of T with respect to t. Start by using the product rule. We have $\displaystyle \frac{1}{4\pi\alpha}t^{-1/2}$ times $\displaystyle \int_{x'= 0}^\infty F'(x')\left(e^{-\frac{(x-x')^2}{4\alpha}t^{-1}+ e^{-\frac{(x+x')^2}{4\alpha}t^{-1}\right)dx'$.

I presume you know that the derivative of $\displaystyle t^{-1/2}$ is $\displaystyle -\frac{1}{2}t^{-3/2}$ so this derivative is $\displaystyle -\frac{1}{8\pi\alpha}t^{-3/2}\int_{x'= 0}^\infty F'(x')\left(e^{-\frac{(x-x')^2}{4\alpha}t^{-1}+ e^{-\frac{(x+x')^2}{4\alpha}t^{-1}\right)dx'$ **plus** $\displaystyle \frac{1}{4\pi\alpha}t^{-1/2}$ times the derivative of that integral with respect to t.

Since t does not appear in either limit of integration, the derivative of the integral is just $\displaystyle \int_{x'= 0}^\infty F'(x')\frac{\partial }{\partial t}\left(e^{-\frac{(x-x')^2}{4\alpha}t^{-1}+ e^{-\frac{(x+x')^2}{4\alpha}t^{-1}\right)dx'$.