Derivatives of Logs and Exponentials

What is the rule for taking the derivative of

 $f\left(x\right)=\log_au$  

•  $f'\left(x\right)=\frac{u}{\ln a}$  

•  $f'\left(x\right)=u'\times u\times\ln a$  

•  $f'\left(x\right)=\frac{u'}{u\ln a}$  

•  $f'\left(x\right)=\frac{1}{u\ln a}$  

What is the rule for taking the derivative of

 $f\left(x\right)=a^u$  

•  $f'\left(x\right)=\frac{u}{\ln a}$  

•  $f'\left(x\right)=u'\cdot u\cdot\ln a$  

•  $f'\left(x\right)=u\ln a$  

•  $f'\left(x\right)=\frac{1}{u\ln a}$  

Find the derivative of 

 $f\left(x\right)=3^{2x^2}$  

•  $f'\left(x\right)=\ln3\cdot2x^2$  

•  $f'\left(x\right)=4x\cdot3^{2x^2}\cdot\ln3$  

•  $f'\left(x\right)=2x^2\cdot3^{x^2}$  

•  $f'\left(x\right)=2x^2\cdot3^{2x^2}\cdot\ln3$  

Find the derivative of

 $f\left(x\right)=\log_5\left(4x\ -\ 3\right)$  

•  $f'\left(x\right)=\frac{4}{\left(4x-3\right)\ln5}$  

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

•  $f'\left(x\right)=\frac{\left(4x-3\right)}{\ln5}$  

•  $f'\left(x\right)=\frac{4}{\ln5}$  

Find the derivative of

 $f\left(x\right)=\log_7\left(\sin x\right)$  

•  $f'\left(x\right)=\frac{\cot x}{\ln7}$  

•  $f'\left(x\right)=\frac{\tan x}{\ln7}$  

•  $f'\left(x\right)=\frac{1}{\sin x\left(\ln7\right)}$  

•  $f'\left(x\right)=\frac{1}{\cos x\left(\ln7\right)}$  

Find the derivative of 

 $f\left(x\right)=5^{\cot x}$  

•  $f'\left(x\right)=5^{\cot x}\ln5$  

•  $f'\left(x\right)=\cot x\cdot5^{\cot x}\cdot\ln5$  

•  $f'\left(x\right)=\cot x\cdot5^{\cot x}$  

•  $f'\left(x\right)=-\csc^2x\cdot5^{\cot x}\cdot\ln5$  

Find the derivative of 

 $f\left(x\right)=\log_9\left(2x^4\right)$  

•  $f'\left(x\right)=\frac{4}{x\ln9}$  

•  $f'\left(x\right)=\frac{8x^3}{\ln9}$  

•  $f'\left(x\right)=\frac{1}{2x^4\ln9}$  

•  $f'\left(x\right)=\frac{2}{x^4\ln9}$  

Find the derivative of 

 $f\left(x\right)=4^{x^2}\cdot\log_45x$  

•  $f'\left(x\right)=2x\cdot4^{x^2}\cdot\ln4+\frac{5}{x\ln4}$  

•  $f'\left(x\right)=\frac{4^{x^2}}{x\ln4}+2x\ln4\cdot4^{x^2}\cdot\log_45x$  

•  $f'\left(x\right)=2x\cdot4^{x^2}\cdot\ln4\cdot\frac{5}{x\ln4}$  

•  $f'\left(x\right)=\frac{4^{x^2}}{x\ln4}$