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

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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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