Ln Differentiation

Ln Differentiation

by Thavarajah Selvarajah

cant recall how to differentiate ln?​

LEVEL 1

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DIRECT

LOG DIFFERENTIATION

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Multiple Select

Given  $$y=\ln\ \left(3x\right).\ \frac{dy}{dx}=?$$  

• $$\frac{1}{3x}$$  

• $$\frac{1}{x}$$  

• $$\frac{3}{3x}$$  

Multiple Choice

$$y=-3\ln\left(x^2+3x-1\right).\ \frac{dy}{dx}=?$$  

• $$\frac{-3}{x^2+3x-1}$$  

• $$\frac{2x+3}{x^2+3x-1}$$  

• $$\frac{-6x-9}{x^2+3x-1}$$  

• $$\frac{1}{x^2+3x-1}$$  

Multiple Choice

$$y=\ln\sqrt[]{2+x}$$  .  $$\frac{dy}{dx}=?$$  

• $$\frac{1}{\sqrt[]{2+x}}$$  

• $$\frac{1}{2\left(2+x\right)}$$  

• $$\sqrt[]{2+x}$$  

• $$\frac{1}{2}\ln\left(2+x\right)^{-\frac{1}{2}}$$  

​LEVEL 2

​LOG DIFFERENTIATION RULES

​AND

​USE OF SEVERAL WEAPONS

Multiple Choice

Given  $$y=\left(\ln\ 2x\right)^3$$  . What is the BEST WEAPON to find $$\frac{dy}{dx}=?$$  

• Power Rule

• Quotient Rule

• Product Rule

• Chain Rule

• Direct

Multiple Choice

Given  $$y=\left(\ln\ 2x\right)^3.\ \frac{dy}{dx}=?$$  

• $$3\left(\ln2x\right)^2$$  

• $$3\left(\ln2x\right)^2\left(\frac{1}{x}\right)$$  

• $$3\left(\ln2x\right)^2\left(\frac{1}{2x}\right)$$  

Multiple Choice

Given $$y=2x\ln\ 3x.$$  What is the BEST WEAPON to find $$\frac{dy}{dx}=?$$  

• Power rule

• Product Rule

• Quotient Rule

• Chain Rule

• Direct

Multiple Choice

Given $$y=2x\ln\ 3x,\ \ \frac{dy}{dx}=?$$  

• $$2\ln3x+2$$  

• $$2\ln3x+\frac{2}{3}$$  

• $$2\left(\frac{1}{x}\right)$$  

• $$2x\ln3x+\frac{2}{x}$$  

Multiple Choice

Given  $$y=\frac{e^{2x}}{\ln x}$$  . What is  $$\frac{dy}{dx}=?$$  

• $$\frac{\frac{e^{2x}}{x}-2e^{2x}\ln x}{\left(\ln x\right)^2}$$  

• $$\frac{2e^{2x}-\frac{e^{2x}}{x}}{\left(\ln x\right)^2}$$  

• $$\frac{2e^{2x}}{\frac{1}{x}}$$  

• $$\frac{e^{2x}-\frac{e^{2x}}{x}}{\left(\ln x\right)^2}$$

​LEVEL 3

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law of log + differentiation 

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Multiple Choice

Given  $$y=\ln\left(5x^3-1\right)^2\left(\sqrt[]{x^2+1}\right)$$  . Which shows the correct use of law of log and ready to obtain $$y'$$  ?

• $$y=\ln\left(5x^3-1\right)^2+\ln\sqrt[]{x^2+1}$$  

• $$y=2\ln\left(5x^3-1\right)^{ }+\ln\sqrt[]{x^2+1}$$

• $$y=2\ln\left(5x^3-1\right)^{ }+\frac{1}{2}\ln\left(x^2+1\right)$$

• $$y=\ln\left(5x^3-1\right)^2+\ln\left(x^2+1\right)^{\frac{1}{2}}$$

Multiple Choice

Given $$y=\ln\left(\frac{\sqrt[]{x-2}}{\left(x^2+2\right)^2}\right)$$  .Which shows the correct use of law of log and READY to obtain $$y'$$  ?

• $$y=\ln\sqrt[]{x-2}-\ln\left(x^2+2\right)^2$$  

• $$y=\ln\sqrt[]{x-2}-2\ln\left(x^2+2\right)^{ }$$

• $$y=\frac{1}{2}\ln\left(x-2\right)-2\ln\left(x^2+2\right)^{ }$$

• $$y=\frac{1}{2}\ln\left(x-2\right)+2\ln\left(x^2+2\right)^{ }$$

Multiple Choice

Given  $$y=\ln\left(\frac{x^2-1}{x^2\left(1-2x^3\right)}\right).$$  Which shows the correct use of law of log and READY to obtain $$y'$$  ?

• $$y=\ln\left(x^2-1\right)-\ln\left[x^2\left(1-2x^3\right)\right]$$

• $$y=\ln\left(x^2-1\right)-\ln\left(x^2\right)+\ln\left(1-2x^3\right)$$

• $$y=\ln\left(x^2-1\right)-\ln\left(x^2\right)-\ln\left(1-2x^3\right)$$

• $$y=\ln\left(x^2-1\right)-2\ln\left(x^{ }\right)-\ln\left(1-2x^3\right)$$

Multiple Choice

Given  $$y=\ln\left(5x^3-1\right)^2\left(\sqrt[]{x^2+1}\right)$$ . What is  $$\frac{dy}{dx}=?$$  

• $$2\left(\frac{15x^2}{5x^3-1}\right)+\frac{1}{2}\left(\frac{2x}{x^2+1}\right)$$  

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

• $$\frac{15x^2}{5x^3-1}+\frac{2x}{x^2+1}$$  

• $$\frac{30x^2}{10x^3-2}+\frac{x}{\frac{1}{2}x^2+\frac{1}{2}}$$  

Multiple Choice

Given $$y=\ln\left(\frac{\sqrt[]{x-2}}{\left(x^2+2\right)^2}\right)$$ . $$\frac{dy}{dx}=?$$  

• $$\frac{1}{x-2}-\frac{1}{x^2+2}$$  

• $$\frac{1}{x-2}-\frac{2x}{x^2+2}$$  

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

• $$\frac{1}{2x-4}-\frac{4x}{2x^2+4}$$  

Multiple Choice

Given  $$y=\ln\left(\frac{x^2-1}{x^2\left(1-2x^3\right)}\right).$$ What is  $$\frac{dy}{dx}=?$$  

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

• $$\frac{2x}{x^2-1}-2\left(\frac{1}{x}\right)-\frac{6x^2}{1-2x^3}$$  

• $$\frac{2x}{x^2-1}-2\left(\frac{1}{x}\right)+\frac{6x^2}{1-2x^3}$$  

• $$\frac{2x}{x^2-1}-\frac{1}{x}+\frac{6x^2}{1-2x^3}$$