The Ultimate Crash Course for STEM Majors

This is nothing more than a free preview/sample of The Ultimate Crash Course for STEM Majors. This lesson reviews exponential and logarithmic formulas, inverse functions, logarithmic derivatives, exponential derivatives, and the derivation of the exponential derivative formula.

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Exponential and Logarithmic Derivatives

Note* $\log x=\ln x$. In a TI calculator, LOG has a base of 10. This is a TI calculator. Anywhere else, $\log x=\ln x$.

$$ \log_e x=\log x, \qquad \log_e x=\frac{\ln x}{\ln e}=\frac{\ln x}{1}=\ln x $$

Formulae

$$ \ln x^a=a\ln x, \qquad \ln xy=\ln x+\ln y, \qquad \ln\frac{x}{y}=\ln x-\ln y, \qquad \ln e^x=e^{\ln x}=x. $$

$$ f^{-1}(f(x))=x=f(f^{-1}(x)) \quad\Leftrightarrow\quad \ln e^x=e^{\ln x}=x \qquad [\text{inverse composition}] $$

$$ \log_a b=\frac{\ln b}{\ln a}. $$

$$ \ln^{-1}x\neq\frac{1}{\ln x}, \qquad \ln x^{-1}=\ln\left(\frac{1}{x}\right), \qquad f^{-1}(x)=\ln^{-1}x, \qquad [f(x)]^{-1}=[\ln(x)]^{-1}=\frac{1}{\ln x}. $$

Derivative of Logarithmic Function

$$ \frac{d}{dx}\log_x x^2 = \frac{d}{dx}\frac{\ln x^2}{\ln x} = \frac{d}{dx}2\frac{\ln x}{\ln x} = 2\frac{d}{dx}\frac{\ln x}{\ln x} = 2\frac{d}{dx}(1) = 2(0) = 0. $$

Derivative of Exponential Functions

$$ \frac{d}{dx}e^u=e^u u’, \qquad \frac{d}{dx}a^u=a^u\ln(a)\,u’, \qquad \frac{d}{dx}\ln u=\frac{u’}{u}. $$

Let’s derive the exponential derivative formula for $a^u$. Let $y=a^u$ and find the derivative.

$$ y=a^u \quad\Rightarrow\quad \ln y=\ln a^u \quad\Rightarrow\quad \ln y=u\ln a $$

$$ \Rightarrow \frac{d}{dx}[\ln y=u\ln a] \Rightarrow \frac{y’}{y}=u’\ln a \Rightarrow y’=u’\ln a\,y = u’\ln a\,a^u. $$

$$ \therefore \qquad y=a^u \quad\Rightarrow\quad y’=a^u\ln(a)\,u’. $$

Exponential Derivative Example

$$ \frac{d}{dx}\pi^{x^2} = \pi^{x^2}\ln(\pi)(2x) = 2x\pi^{x^2}\ln\pi = 2x\pi^{x^2}\log\pi. $$

Continue Learning with The Ultimate Crash Course

This exponential and logarithmic derivatives lesson is nothing more than a free preview/sample of The Ultimate Crash Course for STEM Majors. Continue with the series for additional lessons covering algebra, calculus, derivatives, logarithms, exponential functions, physics, differential equations, and engineering mathematics.

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The Ultimate Crash Course for STEM Majors exponential functions logarithms and derivatives