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Calculus/I need help with a derivative

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Expert: - 2/10/2010


Question
i need to find y'(1) for the following

y = x^3/2 + 2
      ___________
             x

Answer
We have y = f(x)/g(x).

I will assume that x^3/2 is really suppose to be √x³ and not x³/2.

To rewrite the function, it is y = (√x³ + 2)/x.

The quotient rule says that y' = (gf' - fg')/g².

Since f(x) = √x³ + 2, f'(x) = 1.5√x.  Since g(x) = x, g(x) = 1.

Using the formula given, we have y' = (x(1.5√x) - (√x³ + 2)1)/x².

Multiplying out the first term gives (1.5x²√x - x√x - 2)/x².

To find y'(1), put in x = 1.

This gives (1.5 - 1 - 2)/1 = -1.5/1 = -1.5.

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