Calculating Derivative – Square root inside ln with a parameter – Exercise 6269

Exercise

Find the derivative of the following function:

$f(x)=\ln\sqrt{a^2-x^2}$

a is a parameter.

Final Answer

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Solution

$f(x)=\ln\sqrt{a^2-x^2}$

Using Derivative formulas and chain rule in Derivative Rules, we get the derivative:

$f'(x)=\frac{1}{\sqrt{a^2-x^2}}\cdot\frac{1}{2\sqrt{a^2-x^2}}\cdot (-2x)=$

One can simplify the derivative:

$=\frac{1}{2(a^2-x^2)}\cdot (-2x)=$

$=\frac{x}{x^2-a^2}$

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