Ln Differentiation

$$\frac{1}{3x}$$  

$$\frac{1}{x}$$  

$$\frac{3}{3x}$$  

$$\frac{-3}{x^2+3x-1}$$  

$$\frac{2x+3}{x^2+3x-1}$$  

$$\frac{-6x-9}{x^2+3x-1}$$  

$$\frac{1}{x^2+3x-1}$$  

$$\frac{1}{\sqrt[]{2+x}}$$  

$$\frac{1}{2\left(2+x\right)}$$  

$$\sqrt[]{2+x}$$  

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

Power Rule

Quotient Rule

Product Rule

Chain Rule

Direct

$$3\left(\ln2x\right)^2$$  

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

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

Power rule

Product Rule

Quotient Rule

Chain Rule

Direct

$$2\ln3x+2$$  

$$2\ln3x+\frac{2}{3}$$  

$$2\left(\frac{1}{x}\right)$$  

$$2x\ln3x+\frac{2}{x}$$  

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

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

$$\frac{2e^{2x}}{\frac{1}{x}}$$  

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

$$\frac{1}{x-2}-\frac{1}{x^2+2}$$  

$$\frac{1}{x-2}-\frac{2x}{x^2+2}$$  

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

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

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

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

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

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