Differential Chain Rule

In differential calculus, the chain rule is a formula for the derivative of the composition of two functions.In intuitive terms, if a variable, y, depends on a second variable, u, which in turn depends on a third variable, x, that is y = y(u(x)) , then the rate of change of y with respect to x is,(Source : wikipedia)
 `(dy/dx)`   = `(dy/(du))` .`((du )/(dx)) `   

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Problems using differntial chain rule:

Problem 1:
Find the derivative of a function f(x) = ( x³ - 2x + 4)² by using differential chain rule.     
Solution:
Given f(x) = ( x³ - 2x + 4)²
Let us take u = x³ - 2x + 4
f(x) = u²
Differentiate f(x)  with respect to u,
f'(x) = (2u)u'
       = 2( x³ - 2x + 4 )( x³ - 2x + 4)'
      = 2( x³ - 2x + 4) (3x² - 2)
      = 2( 3x⁵ - 6x³ +12x² -2x³ + 4x -8)
      = 2( 3x⁵ - 8x³ +12x²  + 4x -8)
Answer : f'(x) = 2( 3x⁵ - 8x³ +12x²  + 4x -8)

Problem 2:
By using the differential chain rule find the first derivative of the trigonometric function f(x) = (sin x)(cos² x)
Solution:
Given f(x) = (sin x)(cos² x)
Differential f(x) with respect to x,
f'(x) = (sin x) ( 2 cos x) ( -sin x) + (cos² x) (cos x)
       = -2 (sin² x) ( cos x) + cos³ x
       = cos x ( -2 sin² x + cos² x)
Answer : f'(x) = cos x ( -2 sin² x + cos² x)

Problem 3:
Differentiate the function f(x) =` (x^2 - 4) / (x - 2)`  by using the differential chain rule.
Solution:
Given f(x) = ` (x^2 - 4) / (x -2)`
 f(x) = ` (x^2 -4) / (x -2)`
        =  `(x^2 -2^2) / (x-2)`
       =  `((x-2)(x+2)) /(x-2)`
       = x+2
f(x) = x+2
Differentiate f(x) with respect to x,
f'(x) = 1+2
       = 3
Answer : f'(x) =3


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Practice problems on differential chain rule:

1.Find the derivative f(x) = (cos x)(sin x)
2.Find the derivative y = x⁴-5x²
Answer key:
1. -sin² x + cos² x
2. 4x³-10x