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

Exercise

Find the derivative of the following function:

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

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

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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