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Calculating Derivative – Square root in ln function – Exercise 6367

Exercise

Find the derivative of the following function:

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

Final Answer


f'(x)=\frac{\ln x}{2x}

Solution

We simplify the function before differentiating:

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

=\ln^2 x^{\frac{1}{2}}=

={(\frac{1}{2}\ln x)}^2=

=\frac{1}{4}\ln^2 x

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

f'(x)=\frac{1}{4}\cdot 2\ln x\cdot\frac{1}{x}=

One can simplify the derivative:

=\frac{\ln x}{2x}

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