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what's the derivative of f(x) = 4x divided by square root of x^2- 8

Calculus
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what's the derivative of f(x) = 4x - divided by the square root x^2 - 8?

Apr 23rd, 2015

I read the problem as follows: f(x) = 4x / sqrt(x^2 -8)

if it's not correct, let me know. Let me abbreviate f(x) as y, Then y = 4x * sqrt(x^2 -8)^(-1/2). Rewriting the problem in this way allows you to use the product rule instead of the (more complicated, IMHO) quotient rule.

Then fprime(x) = 4x * d(sqrt(x^2-8)^(-1/2)) + sqrt(x^2-8)^(-1/2) * d(4x)/dx. Simplifying, we get

fprime(x) = 4x * d(sqrt(x^2-8)^(-1/2)) + sqrt(x^2-8)^(-1/2) * 4 

You now need to apply the chain rule to differentiate sqrt(x^2-8)^(-1/2). I'll just do this, as follows:

d(sqrt(x^2-8)^(-1/2)) = (-1/2) * (x^2 -8)^(-3/2) * (2x -0), so

dy/dx = -4x^2/(x^2 -8)^(3/2) + 4/(x^2 -8)^(12)



Apr 22nd, 2015

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