problemHow to differentiate y=(x^5+3x^2+cosx)^2

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giorgiana1976 | College Teacher | (Level 3) Valedictorian

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To use the chain rule, we'll specify first that f(x) is the result of composition of 2 functions.

u(x) = x^5 + 3x^2 + cos x and v(u) = u^2

f(x) = (vou)(x) = v(u(x)) = v(x^5 + 3x^2 + cos x) = (x^5 + 3x^2 + cos x)^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^5 + 3x^2 + cos x)' = 5x^4 + 6x - sin x

f'(x) = 2u*(5x^4 + 6x - sin x)

We'll substitute u and we'll get:

The derivative of f(x) is: f'(x) = 2(x^5 + 3x^2 + cos x)*(5x^4 + 6x - sin x)

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