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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


f'(x)=\frac{x}{x^2-a^2}

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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