Ch 4 Year End Review- Derivatives of Logs and Exponentials

Ch 4 End of the year Review

Derivatives of Logs and Exponentials

Example 1

 $Given\ f\left(x\right)=e^{3x^2}find\ f'\left(x\right)$  
1) recopy the exponential piece:    $e^{3x^2}$  

2) multiply by the natural log of the base:  $e^{3x^2}\left(\ln e\right)$  
3)  multiply by the derivative of the exponent and simplify:                                                            $e^{3x^2}\left(\ln e\right)\left(6x\right)=e^{3x^2}\left(1\right)\left(6x\right)=6xe^{3x^2}$  

Multiple Choice

 $Find\ the\ derivative\ of:\ \ f\left(x\right)=e^{5x^3-2x}$   

•  $e^{5x^3-2x}$  

•  $e^{5x^3-2x}\left(\ln5\right)$  

•  $\left(15x^2-2\right)e^{5x^3-2x}$  

•  $\left(\frac{1}{5}e^{5x^3-2x}\right)$  

Multiple Choice

Find y' if y=e^3x

• e^3x

• e^2x

• 0

• 3e^3x

Multiple Choice

Find the derivative:

y=4e^x²

• y'=8xe^x

• y'=8xe^x²

• y'=8xe^2x

• y'=8xe^4x²

Multiple Choice

Find the derivative:

y=e^{4x^2-5x+6}

• e^{4x^2-5x+6}

• e^8x-5

• (8x-5)e^{4x^2-5x+6}

• (4x^2-5x+6)e^{4x^2-5x+6}

Multiple Choice

Evaluate  $\frac{\text{d}}{\text{d}x}\left[2^x\right]$  

•  $2^x$  

•  $\left(\ln2\right)2^x$  

•  $\left(\ln2\right)e^x$  

•  $2\times2^x$  

Multiple Choice

Find the derivative:

y=(2^3x+4)

• 3(ln2)(2^3x+4)

• 3(2^3x+4)

• 3(2^3x+4/(ln2))

• 2^3x+4

Multiple Choice

Find the derivative f(x) = xe^x

• f'(x) = e^x

• f'(x) = xe^x + xe^x

• f'(x) = e^x - xe^x

• f'(x) = e^x + xe^x

Example 2: Derivatives of Logs

 $f\left(x\right)=\ln\left(3x+7\right)$  

1) We write 1 over whatever is inside the log function:  1/(3x+7) 
2) We multiply by the natural log of the base of our log function also in the denominator:   $\frac{1}{\left(3x+7\right)\ln e}$  
3) We multiply all of this by the derivative of what was inside the log:   $\frac{1}{\left(3x+7\right)\ln e}\cdot\ \left(3\right)$  
4) Finally, we simplify.   $\frac{1}{\left(3x+7\right)\left(1\right)}\cdot\ \left(3\right)=\ \frac{3}{3x+7}$  

Multiple Choice

Evaluate  $\frac{\text{d}}{\text{d}x}\left[\ln\ 2x\right]$ 

•  $\frac{2}{x}$  

•  $\frac{1}{2x}$  

•  $\frac{1}{\left(\ln\ 2\right)x}$  

•  $\frac{1}{x}$  

Multiple Choice

Find the derivative:

y = 3ln(x²-3)

• 6x/(x²-3)

• 3/(x²-3)

• 3x/(x²-3)

• 9x/(x²-3)

Multiple Choice

What is the rule for taking the derivative of

 $f\left(x\right)=\log_au$  

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

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

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

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

Multiple Choice

Find the derivative of 

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

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

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

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

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

Multiple Choice

Find the derivative of

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

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

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

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

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

Multiple Choice

Evaluate  $\frac{\text{d}}{\text{d}x}\left[\log_2x\right]$  

•  $\frac{2}{x}$  

•  $\frac{1}{2x}$  

•  $\frac{1}{\left(\ln2\right)x}$  

•  $\frac{1}{\log_2x}$  

Multiple Choice

Find the derivative of 

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

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

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

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

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