Calculating Derivative – Multiplication of a rational function and ln function – Exercise 6339

Exercise 

Find the derivative of the following function:

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

Final Answer

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Solution

We arrange the function before differentiating:

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

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

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

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

One can simplify the derivative:

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

Note: You can also leave the function as a multiplication of two functions and derive it using the multiplication rule in derivation rules. It leads to the same result, of course.

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