Derivatives of Logarithmic Functions

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Derivatives of Logarithmic Functions
The Derivative of the Natural Log Function
The derivative of the natural logarithm function f(x)=ln⁡xf(x)=\ln xf(x)=lnxis

(ln⁡x)′=1x\boxed{\quad (\ln x)'=\frac{1}{x}\quad }(lnx)′=x1​​
The Derivative of General Log Functions
For a given b>0b>0b>0 and b≠1 b\ne 1b=1, the derivative of the logarithmic function log⁡bx\log_bxlogb​x is:
(log⁡bx)′=1xln⁡b\boxed{\quad (\log_{b}x)'=\frac{1}{x\ln b}\quad }(logb​x)′=xlnb1​​

Wize Concept
The Product, Quotient, and Chain Rules still apply for Log Derivatives!

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Example: Log Derivatives
Find the derivative of y=ln⁡(2x2)y=\ln (2x^2)y=ln(2x2)

y′=4x2x2=2x\displaystyle y'=\frac{4x}{2x^2}=\frac{2}{x}y′=2x24x​=x2​

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Example: Log Derivatives
Find dydx\displaystyle
\frac{dy}{dx}dxdy​ for  f(x)=log2(x2+2xex)\displaystyle f(x)=\text{log}_{2}\left(x^2+2xe^x\right)f(x)=log2​(x2+2xex)

f′(x)=2x+2ex+2xexln⁡2(x2+2xex)f'(x)= \dfrac{2x+2e^x+2xe^x}{\ln2(x^2+2xe^x)}f′(x)=ln2(x2+2xex)2x+2ex+2xex​

Find dydt\displaystyle \frac{dy}{dt}dtdy​ for y=5log⁡5(log⁡2t)\displaystyle y=5\log_{5}(\log_{2}t)y=5log5​(log2​t).

y′=5tln⁡tln⁡5y'\displaystyle =\frac{5}{t\ln t\ln5}y′=tlntln55​

y′=5tln⁡tln⁡5y'\displaystyle =5t\ln t\ln5y′=5tlntln5

y′=5ln⁡tln⁡5y'\displaystyle =\frac{5}{\ln t\ln5}y′=lntln55​

y′=1ln⁡tln⁡5y'\displaystyle =\frac{1}{\ln t\ln5}y′=lntln51​

I don't know

Find the derivative of y=log3(7xln⁡2)\displaystyle y=\text{log}_3(7x^{\ln2})y=log3​(7xln2)

ln⁡2xln⁡3\displaystyle \frac{\ln2}{x\ln3}xln3ln2​

17xln⁡2ln⁡3\displaystyle \frac{1}{7x^{\ln2}\ln3}7xln2ln31​

ln⁡2xln⁡2+1ln⁡3\displaystyle \frac{\ln2}{x^{\ln2+1}\ln3}xln2+1ln3ln2​

7ln⁡3xln⁡2\displaystyle \frac{7\ln3}{x\ln2}xln27ln3​

I don't know

Wize AP Calculus (AB) Textbook > Differentiation 2: Composite, Implicit, & Inverse Functions

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Derivatives of Logarithmic Functions

The Derivative of the Natural Log Function

The Derivative of General Log Functions

Example: Log Derivatives

Example: Log Derivatives