# What is the derivative of f(x) = (x^3-(lnx)^2)/(lnx^2)?

Apr 27, 2016

Use quotent rule and chain rule. Answer is:

$f ' \left(x\right) = \frac{3 {x}^{3} \ln {x}^{2} - 2 {\left(\ln x\right)}^{2} - 2 {x}^{3}}{x {\left(\ln {x}^{2}\right)}^{2}}$

This is a simplified version. See Explanation to watch until which point it can be accepted as a derivative.

#### Explanation:

$f \left(x\right) = \frac{{x}^{3} - {\left(\ln x\right)}^{2}}{\ln} {x}^{2}$

$f ' \left(x\right) = \frac{\left({x}^{3} - {\left(\ln x\right)}^{2}\right) ' \cdot \ln {x}^{2} - \left({x}^{3} - {\left(\ln x\right)}^{2}\right) \left(\ln {x}^{2}\right) '}{\ln {x}^{2}} ^ 2$

$f ' \left(x\right) = \frac{\left(3 {x}^{2} - 2 \ln x \cdot \left(\ln x\right) '\right) \cdot \ln {x}^{2} - \left({x}^{3} - {\left(\ln x\right)}^{2}\right) \frac{1}{x} ^ 2 \left({x}^{2}\right) '}{\ln {x}^{2}} ^ 2$

$f ' \left(x\right) = \frac{\left(3 {x}^{2} - 2 \ln x \cdot \frac{1}{x}\right) \cdot \ln {x}^{2} - \left({x}^{3} - {\left(\ln x\right)}^{2}\right) \frac{1}{x} ^ 2 \cdot 2 x}{\ln {x}^{2}} ^ 2$

At this form, it is actually acceptable. But to further simplify it:

$f ' \left(x\right) = \frac{\left(3 {x}^{2} - 2 \ln \frac{x}{x}\right) \cdot \ln {x}^{2} - \left({x}^{3} - {\left(\ln x\right)}^{2}\right) \frac{2}{x}}{\ln {x}^{2}} ^ 2$

$f ' \left(x\right) = \frac{3 {x}^{2} \ln {x}^{2} - 2 \ln \frac{x}{x} \ln {x}^{2} - {x}^{3} \cdot \frac{2}{x} + {\left(\ln x\right)}^{2} \cdot \frac{2}{x}}{\ln {x}^{2}} ^ 2$

$f ' \left(x\right) = \frac{3 {x}^{2} \ln {x}^{2} - 2 \ln \frac{x}{x} \ln {x}^{2} - {x}^{3} \cdot \frac{2}{x} + {\left(\ln x\right)}^{2} \cdot \frac{2}{x}}{\ln {x}^{2}} ^ 2$

$f ' \left(x\right) = \frac{3 {x}^{3} \ln {x}^{2} - 2 \ln x \ln {x}^{2} - {x}^{3} \cdot 2 + {\left(\ln x\right)}^{2} \cdot 2}{x {\left(\ln {x}^{2}\right)}^{2}}$

$f ' \left(x\right) = \frac{3 {x}^{3} \ln {x}^{2} - 4 {\left(\ln x\right)}^{2} - 2 {x}^{3} + 2 {\left(\ln x\right)}^{2}}{x {\left(\ln {x}^{2}\right)}^{2}}$

$f ' \left(x\right) = \frac{3 {x}^{3} \ln {x}^{2} - 2 {\left(\ln x\right)}^{2} - 2 {x}^{3}}{x {\left(\ln {x}^{2}\right)}^{2}}$