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 d y d x 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) 
= 12
  d y d x = 0
   
(b) 
= x4
  d y d x = 4x3

(c) 
= 3x
  d y d x = 3

(d) 
= 5x3
  d y d x = 3(5x2) = 15x2


(e)

y = 1 x = x 1 d y d x = x 1 1 = 1 x 2

(f)

y = 2 x 4 = 2 x 4 d y d x = 4 ( 2 x 4 1 ) = 8 x 5 = 8 x 5

(g)

y = 2 5 x 2 = 2 x 2 5 d y d x = 2 ( 2 x 2 1 5 ) = 4 x 3 5 = 4 5 x 3

(h)

y = 3 x = 3 ( x ) 1 2 d y d x = 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 d y d x = 3 2 ( 4 x 3 2 1 ) = 6 x 1 2 = 6 x

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