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Calculus/Derivative of logarithm or exponential

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Expert: - 3/21/2010


Question
Derivative of logarithm or exponential

If  y=  1/2ln(x^2+5), then dy/dx=

Answer
If that y = 1/(2ln(x²+5)) or = (1/2)ln(x²+5), which can be rewritten as ln(x²+5) / 2?

If it is y = 1/(2f(x)), y' = f'(x)/(2f²(x)).  If it is y = f(x)/2, then y' = f'(x)/2.

The question I will answer, is given f(x) = ln(x²+5), what is f'(x)?
Using this, either of the proceeding questions can be answer.

The derivative of ln(x) = 1/x, so the derivative of ln(g(x)) = g'(x)/g(x).

In this case, g(x) = x² + 5, so g'(x) = 2x.

This means the derivative of ln(x²+5) is 2x/(x²+5).

Using this, whichever problem it was, the solution can be found.

Mathematically, this is (1/2)ln(x²+5), which I write as ln(x²+5)/2
with the denominator at the end so as not to confuse anyone.

Calculus

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