How do you differentiate #xy=pi/6#?

Question

Charles Anctil · Accepted Answer

#dy/dx=-y/x# differentiate #color(blue)"implicitly with respect to x"# Using the #color(blue)"product rule"# to differentiate the term xy #rArr(x.dy/dx)+(y.d/dx(x))=0# #rArrxdy/dx+y=0rArrxdy/dx=-y# #rArrdy/dx=-y/x#

Emily Ammerman · Answer

undefined To differentiate the equation $ xy = \frac{\pi}{6} $ implicitly with respect to $ x $, you can use the product rule, which states that if $ u $ and $ v $ are functions of $ x $, then $(uv)' = u'v + uv'$. Applying this rule, you'll differentiate both sides of the equation with respect to $ x $ and solve for $ \frac{{dy}}{{dx}} $.

Here are the steps:

1. Take the derivative of both sides of the equation with respect to $ x $.
2. Use the product rule on the left side.
3. Solve for $ \frac{{dy}}{{dx}} $.

Differentiating $ xy = \frac{\pi}{6} $ implicitly with respect to $ x $:

1. $ \frac{{d}}{{dx}}(xy) = \frac{{d}}{{dx}}\left(\frac{\pi}{6}\right) $
2. Applying the product rule, we get:
   $ y + x \frac{{dy}}{{dx}} = 0 $
3. Solve for $ \frac{{dy}}{{dx}} $:
   $ \frac{{dy}}{{dx}} = -\frac{{y}}{{x}} $

So, the derivative of $ xy = \frac{\pi}{6} $ with respect to $ x $ is $ -\frac{{y}}{{x}} $.