How do you differentiate f(x)=sec(1/x^3)f(x)=sec(1x3) using the chain rule?

1 Answer
Mia
Nov 29, 2016

f'(x)=-3/x^4sec(1/x^3)tan(1/x^3)

Explanation:

Differentiating this function is determined by applying chain rule.
" "
Chain Rule:
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color (blue)((h(g(x)))'=h'(g(x))xxg'(x))
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Applying the chain rule :
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(sec(1/x^3))'=sec'(1/x^3)xx (1/x^3)'
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Knowing that:
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d/dx(secx)=tanxsecx
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sec'(1/x^3)=tan(1/x^3)xxsec(1/x^3)
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1/x^3" " can be differentiated by either applying quotient or power rule, here I will apply the power rule.
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(1/x^3)'=x^(-3)=-3x^(-4)=-3/x^4
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Therefore,
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(sec(1/x^3))'=sec'(1/x^3)xx (1/x^3)'
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(sec(1/x^3))'=sec(1/x^3)xxtan (1/x^3)xx(-3/x^4)
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Hence,
" "
f'(x)=-3/x^4sec(1/x^3)tan(1/x^3)

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