Math 22 Derivatives of Logarithmic Functions

Derivative of the Natural Logarithmic Function

 Let $u$ be a differentiable function of $x$.
 1.Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\frac {d}{dx}}[\ln x]={\frac {1}{x}}}
 for $x>0$
 
 2.Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\frac {d}{dx}}[\ln u]={\frac {1}{u}}{\frac {du}{dx}}}
 for $u>0$

Exercises Find the derivative of the function

a) Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle f(x)=\ln(7x)}

Solution:
Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle f'(x)={\frac {1}{7x}}(7x)'={\frac {1}{7x}}7={\frac {1}{x}}}

b) Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle f(x)=\ln(x^{8})}

Solution:
Solution 1: Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle f'(x)={\frac {1}{x^{8}}}(x^{8})'={\frac {8x^{7}}{x^{8}}}={\frac {8}{x}}}
Solution 2: $f(x)=\ln(x^{8})=8\ln x$, so Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f'(x)=8\frac{1}{x}=\frac{8}{x}}

c) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)=\ln (4-x^2)}

Solution:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f'(x)=\frac{1}{4-x^2}(4-x^2)'=\frac{1}{4-x^2}(-2x)=\frac{-2x}{4-x^2}}

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