2.5. The Chain Rule http://www.ck12.org
y= [u(x)]n.The rule for differentiating such functions is called theGeneral Power Rule.It is a special case of the Chain Rule.
The General Power Rule
if
y= [u(x)]nthen
dy
dx=n[u(x)]n− (^1) u′(x).
In simpler form, if
y=un
then
y′=nun−^1 ·u′.
Example 2:
What is the slope of the tangent line to the functiony=
√
x^2 − 3 x+2 that passes through pointx=3?
Solution:
We can writey= (x^2 − 3 x+ 2 )^1 /^2 .This example illustrates the point thatncan be any real number including fractions.
Using the General Power Rule,
dy
dx=1
2 (x(^2) − 3 x+ 2 )^12 − (^1) ( 2 x− 3 )
=^12 (x^2 − 3 x+ 2 )−^1 /^2 ( 2 x− 3 )
= 2 √(x^22 x−− 33 x)+ 2
To find the slope of the tangent line, we simply substitutex=3 into the derivative:
dy
dx
∣∣
∣∣
∣x= 3 =2 ( 3 )− 3
2 √ 32 − 3 ( 3 )+ 2 =
3
2 √ 2 =
3
√
2
4.
Example 3: