2.6. Implicit Differentiation http://www.ck12.org
2.6 Implicit Differentiation
Learning Objectives
A student will be able to:
- Find the derivative of variety of functions by using the technique of implicit differentiation.
Consider the equation
2 xy= 1.We want to obtain the derivativedy/dx. One way to do it is to first solve fory,
y= 21 x,and then project the derivative on both sides,
dy
dx=d
dx[ 1
2 x]
= 2 −x^12.There is another way of findingdy/dx. We can directly differentiate both sides:
d
dx[^2 xy] =d
dx[^1 ].Using the Product Rule on the left-hand side,
ydxd[ 2 x]+ 2 xdxd[y] = 0
y[ 2 ]+ 2 xdydx= 0.Solving fordy/dx,
dy
dx=− 2 y
2 x =−y
x.