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Differentiation of Composite Functions (Chain Rule)

If f(x) and g(x) are differentiable functions, then fog is also differentiable and
(fog)′ (x) = f′(g(x))  g′(x)
 
or, 85135.png {fog) (x)} = 85128.png
 
or
 
If y is a function of t and t is a function of x, then 85121.png = 85114.png. This rule is called chain rule. This chain rule can be extended as follows:
 
Let y = f(t), t = φ(z), z = ψ(x), then 85108.png = 85102.pngf′(t). φ′(z ψ′(x)
 
For example, let y = log sin x3 = log t.
 
Putting t = sin x3 = sin zz = x3, we get
85096.png = (1/t cos z  3x2
= (1/sin x3) (cos x3 3x2 = 3x2  cot x3




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