If f(x)= - e^(5x and g(x) = 2x^3 , how do you differentiate f(g(x)) using the chain rule?

1 Answer
Dec 4, 2016

(f(g(x)))'= -30x^2e^(10x^3)

Explanation:

f(g(x)) is a composite function, so differentiating it is
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determined by applying chain rule.
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Chain rule :
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(f(g(x)))'= f'(g(x))xxg'(x)
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Let us compute " "color(blue)(f'(g((x)))
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Knowing the differentiation of e^u:
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(e^u)' = u'e^u
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f'(x)=-5e^(5x)" "
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then
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f'(g(x)))=-5e^(5(g(x)))
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color(blue)(f'(g(x)))=-5e^(5(2x^3))
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color(blue)(f'(g(x))=-5e^(10x^3))
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Differentiation of g(x)
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g'(x) is determined by applying the power rule.
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Power Rule:
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color(red)((x^n)' = nx^(n-1))
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g'(x)=2(x^3)'=2(color(red)(3x^2))=6x^2
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Therefore ,
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(f(g(x)))'= f'(g(x))xxg'(x)
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(f(g(x)))'= -5e^(10x^3)xx6x^2
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(f(g(x)))'= -30x^2e^(10x^3)