http://www.ck12.org Chapter 2. Derivatives
But sincey= 21 x, substitution gives
dy
dx=− 1
x( 2 x)
= 2 −x^12.which agrees with the previous calculations. This second method is called theimplicit differentiationmethod. You
may wonder and say that the first method is easier and faster and there is no reason for the second method. That’s
probably true, but consider this function:
3 y^2 −cosy=x^3.How would you solve fory? That would be a difficult task. So the method of implicit differentiation sometimes is
very useful, especially when it is inconvenient or impossible to solve foryin terms ofx. Explicitly defined functions
may be written with a direct relationship between two variables with clear independent and dependent variables.
Implicitly defined functions or relations connect the variables in a way that makes it impossible to separate the
variables into a simple input output relationship. More notes on explicit and implicit functions can be found at http
://en.wikipedia.org/wiki/Implicit_function.
Example 1:
Finddy/dxif 3y^2 −cosy=x^3.
Solution:
Differentiating both sides with respect toxand then solving fordy/dx,
d
dx[^3 y(^2) −cosy] = d
dx[x
(^3) ]
(^3) dxd[y^2 ]−dxd[cosy] = 3 x^2
3 ( 2 ydydx)−(−siny)dydx= 3 x^2
6 ydydx+sinydydx= 3 x^2
y + sinydydx= 3 x^2.
Solving fordy/dx, we finally obtain
dy
dx=
3 x^2
6 y+siny.
Implicit differentiation can be used to calculate the slope of the tangent line as the example below shows.
Example 2:
Find the equation of the tangent line that passes through point( 1 , 2 )to the graph of 8y^3 +x^2 y−x= 3.
Solution: