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Wize University Calculus 1 Textbook > Derivatives

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 xis

(ln⁡x)′=1x\boxed{\quad (\ln x)'=\frac{1}{x}\quad }

The Derivative of General Log Functions

For a given b>0b>0 and b≠1 b\ne 1, the derivative of the logarithmic function log⁡bx\log_bx is:
(log⁡bx)′=1xln⁡b\boxed{\quad (\log_{b}x)'=\frac{1}{x\ln b}\quad }

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′=4x2x2=2x\displaystyle y'=\frac{4x}{2x^2}=\frac{2}{x}
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Example: Log Derivatives

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


f′(x)=2x+2ex+2xexln⁡2(x2+2xex)f'(x)= \dfrac{2x+2e^x+2xe^x}{\ln2(x^2+2xe^x)}
Find dydt\displaystyle \frac{dy}{dt} for y=5log⁡5(log⁡2t)\displaystyle y=5\log_{5}(\log_{2}t).
Find the derivative of y=log3(7xln⁡2)\displaystyle y=\text{log}_3(7x^{\ln2})

Extra Practice
Derivatives: Logarithmic Functions
Find the derivative of
f(x)=exx2+ln⁡(x)3f(x) = e^{x} x^2 + \frac{\ln(x)}{3}
Practice: Chain Rule
Find the derivative of y=ln⁡(arctan⁡x)y=\ln(\arctan x)
Derivatives: Exponential and Logarithmic Functions
Find f′(x)f'(x) if f(x)=ln⁡xxex\displaystyle f\left(x\right)=\frac{\ln x}{xe^{x}}. Simplify.
Inverse Trig with Log
Compute the derivative of f(x)=arctan(log10x)\displaystyle f(x)=\text{arctan}\left(\text{log}_{10}x\right)
Applying Rules of Logs
Find dydx\displaystyle\frac{dy}{dx} given y=log⁡6(x−1)5(x+1)103\displaystyle y=\log_6 \sqrt[3]{\frac{(x-1)^{5}}{(x+1)^{10}}}
Practice: Log and Ln
Q.\textbf{Q.} Find the derivative of y=log3(7xln⁡2)\displaystyle y=\text{log}_3(7x^{\ln2})
Derivatives of Logarithmic Functions: Two Logs
Find dy/dxdy/dx for y=5log⁡5(log⁡2t)\displaystyle y=5\log_{5}(\log_{2}t)
Derivatives: Logarithmic Functions
Compute the derivative of f(x)=xx+1f(x) = x^{x + 1}. Remember that log⁡x=log⁡ex=ln⁡x\log x = \log_e x = \ln x.
The chain rule: Logarithmic Derivatives
The derivative of log⁡3(e3x)\log_3\left(e^{3x}\right) is
Derivatives: Logarithmic Functions
Calculate the derivative of the following functions.
f(x)=ln⁡xx+xe2x\displaystyle f(x)=\frac{\ln{x}}{x+xe^{2x}}
Calculate the derivative of the following
f(x)=ln⁡xx+xe2x\displaystyle f(x)=\frac{\ln{x}}{x+xe^{2x}}
Derivatives: Logarithmic Functions
Find the derivative of the following function.
f(x)=ln⁡(x2+x)\displaystyle f(x)=\ln{(x^2+\sqrt{x})}
Derivatives: Logarithmic Functions
Compute the derivative of f(x)=log⁡2(x2+2xex)f(x)=\log_2(x^2+2xe^x)
Differentiate the following:
y=ex2+ln⁡xx\begin{aligned} &y=\frac{e^{x^2+\ln x}}{x} \end{aligned}
Derivatives: Exponential and Logarithmic Functions
Find f′(x)f'(x) if f(x)=ln⁡xxex\displaystyle f\left(x\right)=\frac{\ln x}{xe^{x}}. Simplify.
Inverse Trig with Log
Compute the derivative of f(x)=arctan(log10x)\displaystyle f(x)=\text{arctan}\left(\text{log}_{10}x\right)
Applying Rules of Logs
Find dydx\displaystyle\frac{dy}{dx} given y=log⁡6(x−1)5(x+1)103\displaystyle y=\log_6 \sqrt[3]{\frac{(x-1)^{5}}{(x+1)^{10}}}
Derivatives of Logarithmic Functions: Two Logs
Find dy/dxdy/dx for y=5log⁡5(log⁡2t)\displaystyle y=5\log_{5}(\log_{2}t)
Practice: Log and Ln
Q.\textbf{Q.} Find the derivative of y=log3(7xln⁡2)\displaystyle y=\text{log}_3(7x^{\ln2})
Derivatives: Logarithmic and Inverse Trigonometric Functions
Evaluate ddx(sin⁡−1xsin⁡xlog⁡3x)\displaystyle \frac{\text{d}}{\text{d}x}\left( \frac{\sin^{-1}x\sin x}{\log_3x}\right).
Derivatives: Trigonometric and Logarithmic Functions
Find ddy[(y5+ln⁡y)tan⁡y]\displaystyle \frac{\text{d}}{\text{d}y}\left[\left(y^5+\ln y\right)\tan y\right].
Compute the derivative of f(x)=log⁡2(x2+2xex)f(x) = \log_2(x^2 + 2xe^x).
Derivatives of Logarithmic Functions
Evaluate ddx(4xlog⁡700x)\frac{\text{d}}{\text{d}x}\left(4^x\log_{700}x\right). It is not necessary to simplify your answer.
Find the derivative of the following function.
f(x)=ln⁡(x2+x)\displaystyle f(x)=\ln{(x^2+\sqrt{x})}
Given that f(x)=ecos⁡(ln⁡x)f\left(x\right)=e^{\cos\left(\ln x\right)}, find f′(1)f'\left(1\right).
Practice: Chain Rule
Find the derivative of y=ln⁡(arctan⁡x)y=\ln(\arctan x)
Derivatives: Logarithmic Functions
Find the derivative of
f(x)=exx2+ln⁡(x)3f(x) = e^{x} x^2 + \frac{\ln(x)}{3}
Derivatives: Logarithmic Functions
Find all the points for which the tangent line of the function
f(x)=4ln⁡(x)+x2−3x+5f(x)= 4 \ln (x) + x^2 - 3x + 5
has slope equal to 3.
Derivatives of Logarithmic Functions: Normal Lines
Find the equation of the normal line to f(x)=ln⁡x at the point (e,1)f(x)=\ln{x} \text{ at the point } (e,1)
Note: This questions requires knowledge of derivatives of logarithms
Derivatives: Logarithmic Functions
Find the equation of the tangent line to the given point for the following function.
f(x)=ln⁡(x+3x+1)at x=1f(x)=\ln\left(\frac{x+3}{x+1}\right) \text{at}\ x=1
Derivatives: Exponential and Logarithmic Functions
Find the derivative of the following function.
f(x)=(x2+xln⁡x)exf(x)=(x^2+x\ln{x})e^{\sqrt{x}}