9.5 First Derivatives of Composite Function

(A) Differentiate Composite Function using Chain Rule


Example:
Differentiate y = (x2 – 1)8 .

Solution:


(B) Differentiate Composite Function using Alternative Method
       - Easy Version

Example:
Differentiate y = (x2 – 1)8 .

Solution:
y= ( x 2 1 ) 8 dy dx =8 ( x 2 1 ) 7 d dx ( x 2 1 ) dy dx =8 ( x 2 1 ) 7 ( 2x ) dy dx =16x ( x 2 1 ) 7

Practice 1:
Given that y= 1 3x7 , find  dy dx
Solution:
y= 1 3x7 = ( 3x7 ) 1 dy dx =1 ( 3x7 ) 2 .3 dy dx = 3 ( 3x7 ) 2

Practice 2:
Given that y= 2 x 2 5x+1 , find  dy dx
Solution:
y= 2 x 2 5x+1 = ( 2 x 2 5x+1 ) 1 2 dy dx = 1 2 ( 2 x 2 5x+1 ) 1 2 ( 4x5 ) dy dx = 4x5 2 2 x 2 5x+1