Differentiation using LOG

Differentiation using LOG

 $\log_xy=f\left(x\right)$  

Multiple Choice

Find

 $\frac{\text{d}y}{\text{d}x}\ if\ y=2^{x\log x}$  

•  $2^{x\log x}\ \log2$  

•  $2^{x\log x}\ \left(1+\log x\right)$  

•  $2^{x\log x}\ \left(1+\log x\right)\log2\$  

• none of the above

Multiple Choice

• a

• b

• c

• d

• e

Multiple Choice

• a

• b

• c

• d

• e

Multiple Choice

Find the differentiation of  $\ln\left(\sqrt{x^3+2x}\right)$  

•  $\frac{3x^2+2}{x^3+2x}$  

•  $\left(\frac{1}{2}.\frac{3x^2+2}{x^3+2x}\right)$  

•  $\frac{3x^2+2}{x^3+2}$  

Multiple Choice

Solving by f'(x)=

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

• $f"\left(x\right)=\ln\ \left(2x+\frac{3}{x+1}\right)e^{2x\ }\left(x+1\right)^3$

Fill in the Blanks

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

Fill in the Blanks

 $f\left(x\right)=e^{2x}\left(x+1\right)^3$  
 $Applying\ Ln\ :$  
 $\ln f\left(x\right)=$  

 $f\left(x\right)=e^{2x}\left(x+1\right)^3$  

 $f'\left(x\right)=?$  

Properties of Logs

Ln(x2*e2x)=Lnx2+2xLne=2lnx+2x
ln  $\frac{e^{2x}}{x}$  =Ln  $e^{2x}$  -l n x =2x-Lnx

Derivatives of a Log

y=ln f(x)
dy/dx= f'(x)/f(x)

f(x)= $e^{2x}$  (x-1)

Ln f(x)= 2x + ln (x-1)
D(lnf(x))=D(2x+ln(x-1)
 $\frac{f'\left(x\right)}{f\left(x\right)}$  = $2\ +\frac{1}{x-1}$  

 $f'\left(x\right)=f\left(x\right)\left(2+\frac{1}{x-1}\right)$