When to use chain rule of differentiation ? Please give an example.f(x)=(x^3+4)^4 or f(x) = (x^2+sinx)^2

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giorgiana1976 | College Teacher | (Level 3) Valedictorian

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Use chain rule of differentiation and find derivative of f(x)=(x^3+4)^4?

We'll use the chain rule of differentiation to find the derivative of a function, that is the result of composition of 2 or more functions.

We'll apply chain rule to function f(x) = (x^2+sinx)^2

u(x) = x^2+sinx and v(u) = u^2

f(x) = (vou)(x) = v(u(x)) = v(x^2+sinx) = (x^2+sinx)^2

We'll differentiate f(x) and we'll get:

f'(x) = v'(u(x))*u'(x)

First, we'll differentiate v with respect to u:

v'(u) = 2u^(2-1) = 2u

Second, we'll differentiate u with respect to x:

u'(x) = (x^2+sin x)' = 2x + cos x

f'(x) = 2u*(2x + cos x)

We'll substitute u and we'll get:

The derivative of f(x) is: f'(x) = 2(x^2+sinx)*(2x + cos x)

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