9.2 First Derivative for Polynomial Function

9.2 First Derivative for Polynomial Function



(A) Differentiating a Constant



(B) Differentiating Variable with Index n




(C) Differentiating a Linear Function



(D) Differentiating a Polynomial Function



(E) Differentiating Fractional Function



(F) Differentiating Square Root Function




Example:
Find dy dx for each of the following functions:
(a) y = 12
(b) y = x4
(c) y = 3x
(d) y = 5x3
(e) y= 1 x (f) y= 2 x 4 (g) y= 2 5 x 2 (h) y=3 x (i) y=4 x 3

Solution:
(a) y = 12
      dy dx =0
   
(b) y = x4
      dy dx = 4x3

(c) y = 3x
      dy dx = 3

(d) y = 5x3
      dy dx = 3(5x2) = 15x2

(e)
y= 1 x = x 1 dy dx = x 11 = 1 x 2 (f)
y= 2 x 4 =2 x 4 dy dx =4( 2 x 41 )=8 x 5 = 8 x 5 (g)
y= 2 5 x 2 = 2 x 2 5 dy dx =2( 2 x 21 5 )= 4 x 3 5 = 4 5 x 3 (h)
y=3 x =3 ( x ) 1 2 dy dx = 1 2 ( 3 x 1 2 1 )= 3 2 x 1 2 = 3 2 x (i)
y=4 x 3 =4 ( x 3 ) 1 2 =4 x 3 2 dy dx = 3 2 ( 4 x 3 2 1 )=6 x 1 2 =6 x