Natural Logarithmic Derivatives

Differentiate the function below with respect to x:

 $y=\ln\left(5x^3\right)$  

•  $\frac{dy}{dx}=\frac{1}{5x^3}$  

•  $\frac{dy}{dx}=\frac{1}{15x^2}$  

•  $\frac{dy}{dx}=\frac{3}{x}$  

•  $\frac{dy}{dx}=5x^3$  

Differentiate the function below with respect to x:

 $f\left(x\right)=\ln\left(x^4\right)$  

•  $f'\left(x\right)=\frac{4}{x}$  

•  $f'\left(x\right)=\frac{1}{4x^3}$  

•  $f'\left(x\right)=\frac{1}{x^4}$  

•  $f'\left(x\right)=x^4$  

Differentiate the function below with respect to x:

 $f\left(x\right)=\ln\left(4x^5\right)$  

•  $f'\left(x\right)=\frac{5}{x}$  

•  $f'\left(x\right)=\frac{1}{20x^4}$  

•  $f'\left(x\right)=\frac{1}{4x^5}$  

•  $f'\left(x\right)=4x^5$  

Differentiate the function below with respect to x:

 $f\left(x\right)=\ln\left(3x^2\right)$  

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

•  $f'\left(x\right)=3x^2$  

•  $f'\left(x\right)=\frac{2}{x}$  

•  $f'\left(x\right)=\frac{1}{6x}$  

Differentiate the function below with respect to x:

 $y=\ln\left(3x^3\right)$  

•  $\frac{dy}{dx}=\frac{1}{9x^2}$  

•  $\frac{dy}{dx}=3x^2$  

•  $\frac{dy}{dx}=\frac{1}{3x^2}$  

•  $\frac{dy}{dx}=\frac{3}{x}$