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Calculus/Derivatives - Quotient Rule (Homework Help)
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Question
Hi there,
I just have a question from my textbook that I'm having trouble with.
Find the derivative of:
f(x)= (x^2) / ((√x)-3)
I'm not sure if I change the root sign to x^(1/2) or if I keep it the same. I keep getting weird answers. I understand that the product rule is low D high minus high D low, draw a line, square below.
When f(x) = hi/low, the derivative is (low d hi - hi d low)/low².
I remember this equation as the one with '-Heidi' in it by the "-hi d " in "- hi d low".
Note that I will use h and h(x) to be the same, and g and g(x) to be the same.
That is, if f = g/h, then f' = (hg' - gh')/h².
Since h(x) = x², h'(x) = 2x.
Since g(x) = √x - 3, g'(x) = 1/(2√x).
This makes (hg' - gh')/h² be [x²/(2√x) - (√x - 3)2x]/(x²)².
Note that x²/(2√x) = x^(3/2)/2 and (√x - 3)2x = 2x^(5/2) - 6x.
This makes our equation be (x^(3/2)/2 - 2x^(5/2) + 6x)/x^4.
This can be rewritten as x^(3/2)/(2x^4) - 2x^(5/2)/x^4 + 6x/x^4.
That is the same as 1/(2x^(5/2) - 2/x^(3/2) + 6/x^3.
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