How to simplify a complicated partial differentiation

Question

Given $u = (t^{-1/2})\exp\left[{\frac{-x^2}{4k^2t}}\right]$, what is the best way to differentiate $u$ with respect to $t$, as well as $u$ with respect to $x$? I am having a very difficult time trying to find a less tedious way of going about this problem.

Answer 1

Use the product rule and chain rule for $\dfrac{d}{dt}$:

$$\large \dfrac{d}{dt} \left((t^{-1/2})\exp\left[{\frac{-x^2}{4k^2t}}\right]\right) = \frac{x^2 e^{-\frac{x^2}{4 k t^2}}}{2 k t^{7/2}}-\frac{e^{-\frac{x^2}{4 k t^2}}}{2 t^{3/2}}$$

For $\dfrac{d}{dx}$, it is just the derivative of the exponential with the chain rule:

$$ \large \dfrac{d}{dx} \left((t^{-1/2})\exp\left[{\frac{-x^2}{4k^2t}}\right]\right) = -\frac{x e^{-\frac{x^2}{4 k t^2}}}{2 k t^{5/2}}$$

Answer 2

Some amount of tedium is inevitable. My laptop battery is about to die. For the $t$ derivative, you could define $$r \equiv \left({4k^2\over x^2}\right)\,t\,,$$ then the function is $$\left({x\over 2k}\right)\,r^{-{1\over 2}}e^{-{1\over r}}\,.$$ Derivatives with respect to $r$ are cleaner than with respect to $t$. Note $${\partial u\over\partial t} = {\partial r\over\partial t} {\partial u \over \partial r} = \left({4k^2\over x^2}\right) {\partial u\over\partial r}\,.$$

I am assuming you meant partial derivatives.

Answer 3

Hint

When you have a rather complex product of terms, a rather simple solution consists in taking the logarithms of both sides and perform differentiation. This will partly eliminate the need of the chain rule. $$u = (t^{-1/2})\exp\left[{\frac{-x^2}{4k^2t}}\right]$$ Then $$\log(u)=-\frac{1}{2}\log (t)-\frac{x^2}{4 k^2 t}$$ So, differentiating $$\frac{u'_x}{u}=-\frac{x}{2 k^2 t}$$ $$\frac{u'_t}{u}=-\frac{1}{2 t}+\frac{x^2}{4 k^2 t^2}$$