# Thread: implicit differentiation problem

1. ## implicit differentiation problem

If and , find by implicit differentiation.

2. Originally Posted by tbenne3
If and , find by implicit differentiation.
$\displaystyle \frac{1}{x} + \frac{1}{y} = 2$

$\displaystyle x^{-1} + y^{-1} = 2$

$\displaystyle \frac{d}{dx}(x^{-1} + y^{-1}) = \frac{d}{dx}(2)$

$\displaystyle -x^{-2} + \frac{d}{dy}(y^{-1})\,\frac{dy}{dx} = 0$

$\displaystyle -\frac{1}{x^2} - \frac{1}{y^2}\,\frac{dy}{dx} = 0$

$\displaystyle -\frac{1}{y^2}\,\frac{dy}{dx} = \frac{1}{x^2}$

$\displaystyle \frac{dy}{dx} = -\frac{y^2}{x^2}$.

You also know that when $\displaystyle x = 4, y = \frac{4}{7}$.

So $\displaystyle \frac{dy}{dx}|_{x = 4} = -\frac{\left(\frac{4}{7}\right)^2}{4^2}$

$\displaystyle = -\frac{\frac{16}{49}}{16}$

$\displaystyle = -\frac{1}{49}$.

3. thanks but one question.. in step 3 how did you get the -x^2?

4. Originally Posted by tbenne3
thanks but one question.. in step 3 how did you get the -x^2?
What's the derivative of $\displaystyle x^{-1}$?