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Question
Prepping for ACTs, having trouble with this question. Working on the Chain Rule.

If f(x)=sqrt(x^4-4x^2), find f'(x)

Thank-You

Answer
Hi John,
The simplest way to work on the chain rule is to rewrite the expression in the following manner;
y = sqrt(x^4 - 4x^2)
Let u = x^4 - 4x^2
then y = sqrt(u) = u^(1/2)
and so we have;
du/dx = 4x^3 - 8x
dy/du = (1/2)u^(-1/2) = 1/[2sqrt(u)]

Now, we can employ the formula
dy/dx = (dy/du) . (du/dx)
= 1/[2sqrt(u)] . (4x^3 - 8x)
= (4x^3 - 8x) / 2sqrt(x^4 - 4x^2)
= (2x^3 - 4x)/sqrt(x^4 - 4x^2)

The expression can still be simplified further to give
dy/dx = (2x^3 - 4x)/ [sqrt(x^2 - 4) . sqrt(x^2)]
= 2x(x^2 - 2) / x(sqrt(x^2 - 4))
= 2(x^2 - 2) / sqrt(x^2 - 4)

Regards

Calculus

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