How do you implicitly differentiate 11=(x-y)/(e^y-e^x)?

Jan 30, 2017

Multiply both sides of the equation by ${e}^{y} - {e}^{x}$
Separate terms containing y to the left and x to the right.
Differentiate each term.
Divide both sides by the coefficient of $\frac{\mathrm{dy}}{\mathrm{dx}}$

Explanation:

Multiply both sides by ${e}^{y} - {e}^{x}$

$11 {e}^{y} - 11 {e}^{x} = x - y$

Add 11e^x + y to both sides:

$y + 11 {e}^{y} = 11 {e}^{x} + x$

Differentiate each term with respect to x:

$\frac{d \left(y\right)}{\mathrm{dx}} + \frac{d \left(11 {e}^{y}\right)}{\mathrm{dx}} = \frac{d \left(11 {e}^{x}\right)}{\mathrm{dx}} + \frac{d \left(x\right)}{\mathrm{dx}}$

The derivative of y is $\frac{\mathrm{dy}}{\mathrm{dx}}$:

$\frac{\mathrm{dy}}{\mathrm{dx}} + \frac{d \left(11 {e}^{y}\right)}{\mathrm{dx}} = \frac{d \left(11 {e}^{x}\right)}{\mathrm{dx}} + \frac{d \left(x\right)}{\mathrm{dx}}$

The derivative of $11 {e}^{y}$ is $11 {e}^{y} \frac{\mathrm{dy}}{\mathrm{dx}}$

$\frac{\mathrm{dy}}{\mathrm{dx}} + 11 {e}^{y} \frac{\mathrm{dy}}{\mathrm{dx}} = \frac{d \left(11 {e}^{x}\right)}{\mathrm{dx}} + \frac{d \left(x\right)}{\mathrm{dx}}$

The derivative of $11 {e}^{x}$ is itself.

$\frac{\mathrm{dy}}{\mathrm{dx}} + 11 {e}^{y} \frac{\mathrm{dy}}{\mathrm{dx}} = 11 {e}^{x} + \frac{d \left(x\right)}{\mathrm{dx}}$

The derivative of x is 1:

$\frac{\mathrm{dy}}{\mathrm{dx}} + 11 {e}^{y} \frac{\mathrm{dy}}{\mathrm{dx}} = 11 {e}^{x} + 1$

Factor dy/dx from the left side:

$\frac{\mathrm{dy}}{\mathrm{dx}} \left(1 + 11 {e}^{y}\right) = 11 {e}^{x} + 1$

Divide both sides $\left(1 + 11 {e}^{y}\right)$

$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{11 {e}^{x} + 1}{1 + 11 {e}^{y}}$