PDE Calculus 3 information

Question

I must compute $ \frac{\partial z}{\partial x} $ of the following expression
$$ z^2 e^{2x} = 3z + x\ln (y) $$

I know how to compute partial derivatives in general , though I'm not sure what to do in this problem , generally what we've covered in class are expressions of the form $$ z = (\dots) $$ then we were asked to compute $ \frac{\partial z}{\partial x} $ or $ \frac{\partial z}{\partial y} $

So from my understanding , I must regroup the $z$ values on one side of the equation and the other variables on the other side , then apply the formulae. Although, I can't seem to do that in this problem ; Attempt :

$$ \frac{x\ln(y)}{e^{2x}-\dfrac{3}{z}}= z^2$$

Also does anyone know what's the command for tilted fractions in MathJax ?

Answer 1

Take implicit derivative of the both sides $ z^2 e^{2x} = 3z + x\ln y $ to get

$$2z\frac{\partial z}{\partial x}e^{2x} +z^2 2e^{2x}=3\frac{\partial z}{\partial x}+ \ln y$$

Thus,

$$\frac{\partial z}{\partial x} = \frac{\ln y - 2z^2 e^{2x} }{2ze^{2x} -3}$$

Similarly,

$$\frac{\partial z}{\partial y} = \frac{x}{y(2ze^{2x} -3)}$$

Answer 2

If having all of the extra stuff on one side really concerns you, you could always rearrange your function like this:

$z = \sqrt{\frac{3z + x \ln(y)}{e^{2x}}}$

Then differentiate with respect to $x$ and $y$.

Do any of these tricks seem to help you with MathJax?

Here are the solutions both with respect to $x$ and $y$, so you can check your work: