# How do you implicitly differentiate 7xy- 3 lny= 42?

Feb 18, 2017

$\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right) = \frac{- 7 y}{\left(7 x\right) - \left(\frac{3}{y}\right)}$

#### Explanation:

Ignore the $- 3 \ln \left(y\right)$ for now. Take the derivative of 7xy with product rule.

$\left(7 x\right) \left(\frac{\mathrm{dy}}{\mathrm{dx}}\right) + \left(y\right) \left(7\right)$

$\left(7 x\right) \left(\frac{\mathrm{dy}}{\mathrm{dx}}\right) + 7 y$

Now take a look at -3lny. Since 3 is a constant in front, you only need to worry about the derivaitve of lny and then multiply the 3 later.

$\ln \left(y\right) = \left(\frac{1}{y}\right) \left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)$

The derivative of a constant is always 0 so the equation will be equal to 0. Now combine your two parts together.

$\left(7 x\right) \left(\frac{\mathrm{dy}}{\mathrm{dx}}\right) + 7 y - \left(\frac{3}{y}\right) \left(\frac{\mathrm{dy}}{\mathrm{dx}}\right) = 0$

Subtract 7y on both sides.

$\left(7 x\right) \left(\frac{\mathrm{dy}}{\mathrm{dx}}\right) - \left(\frac{3}{y}\right) \left(\frac{\mathrm{dy}}{\mathrm{dx}}\right) = - 7 y$

Then factor out $\frac{\mathrm{dy}}{\mathrm{dx}}$ on the left side.

$\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right) \left(\left(7 x\right) - \left(\frac{3}{y}\right)\right) = - 7 y$

Isolate $\frac{\mathrm{dy}}{\mathrm{dx}}$.

$\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right) = \frac{- 7 y}{\left(7 x\right) - \left(\frac{3}{y}\right)}$