Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

How to differentiate $$ \exp \left( -\frac{x^2}{4Dt} \right)$$ ?

(A step by step solution is sought)

share|cite|improve this question
5  
I see that this is related to your previous question where both $x$ and $t$ are variables. So do you want to differentiate with respect to $t$ or $x$ or both? – Srivatsan Nov 21 '11 at 17:06
    
@Srivatsan I wanted to differentiate $P$ with respect to $t$ for the LHS of the equation in the previous question, then $P$ with respect to $x$ twice for the RHS. – ptrcao Nov 22 '11 at 4:37
up vote 4 down vote accepted

Make the simple substitution $u=-\frac{x^2}{4Dt}=-\frac{1}{4}x^2D^{-1}t^{-1}$ and let $z=\mathrm{exp}(u)$. Then...

$$\frac{\partial z}{\partial x} = \frac{dz}{du}\frac{\partial u}{\partial x} = \mathrm{exp}(u) \cdot \left(-\frac{1}{2}xD^{-1}t^{-1} \right) = -\frac{x\cdot \mathrm{exp}\left(-\frac{x^2}{4Dt}\right)}{2Dt}$$

$$\frac{\partial z}{\partial t} = \frac{dz}{du}\frac{\partial u}{\partial t} = \mathrm{exp}(u) \cdot \left(\frac{1}{4}x^2D^{-1}t^{-2} \right) = \frac{x^2\cdot \mathrm{exp}\left(-\frac{x^2}{4Dt}\right)}{4Dt^2}$$

If $D$ is meant to be a variable, we'd also have...

$$\frac{\partial z}{\partial D} = \frac{dz}{du}\frac{\partial u}{\partial D} = \mathrm{exp}(u) \cdot \left(\frac{1}{4}x^2D^{-2}t^{-1} \right) = \frac{x^2\cdot \mathrm{exp}\left(-\frac{x^2}{4Dt}\right)}{4D^2t}$$

Now what if $D$ is constant, $x$ is a function of $t$? Then we have...

$$\frac{dz}{dt} = \frac{dz}{du}\frac{\partial u}{\partial x}\frac{dx}{dt} + \frac{dz}{du}\frac{\partial u}{\partial t}\frac{dt}{dt} = -\frac{x\cdot \mathrm{exp}\left(-\frac{x^2}{4Dt}\right)}{2Dt}\cdot x'(t) + \frac{x^2\cdot \mathrm{exp}\left(-\frac{x^2}{4Dt}\right)}{4Dt^2}\cdot 1$$

Assumptions about dependencies among variables (and what is a variable) can make a big difference.

share|cite|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.