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Notes
Exponential Derivatives
\( \frac{d}{dx} \left(e^x\right)=e^x\)
\( \frac{d}{dx} \left(a^x\right)=a^x \cdot \ln{a}\)
Logarithmic Derivatives
\( \frac{d}{dx} \left(\ln{x}\right)=\displaystyle\frac{1}{x}\)
\( \frac{d}{dx} \left(\log_a{x} \right)=\frac{1}{x \ln{a}}\)
Practice Problems
Find the derivative
\(\textbf{1)}\) \(f(x)=\displaystyle2e^x\)
\(\textbf{2)}\) \(f(x)=\displaystyle3^x\)
\(\textbf{3)}\) \(f(x)=\displaystyle 2e^x+3^x\)
\(\textbf{4)}\) \(f(x)=\displaystyle\frac{1}{e^x}\)
\(\textbf{5)}\) \(f(x)=\displaystyle \frac{1}{3^x}\)
\(\textbf{6)}\) \(f(x)=\displaystyle e^x \ln x\)
\(\textbf{7)}\) \(f(x)=\displaystyle e^x + \ln x\)
\(\textbf{8)}\) \(f(x)=\displaystyle e^{3x}\)
\(\textbf{9)}\) \(f(x)=\displaystyle\pi^x\)
\(\textbf{10)}\) \(f(x)=\displaystyle x^{\pi}\)
See Related Pages\(\)
\(\bullet\text{ Calculus Homepage}\)
\(\,\,\,\,\,\,\,\,\text{All the Best Topics…}\)
\(\bullet\text{ Definition of Derivative}\)
\(\,\,\,\,\,\,\,\, \displaystyle \lim_{\Delta x\to 0} \frac{f(x+ \Delta x)-f(x)}{\Delta x} \)
\(\bullet\text{ Equation of the Tangent Line}\)
\(\,\,\,\,\,\,\,\,f(x)=x^3+3x^2−x \text{ at the point } (2,18)\)
\(\bullet\text{ Derivatives- Constant Rule}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{d}{dx}(c)=0\)
\(\bullet\text{ Derivatives- Power Rule}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{d}{dx}(x^n)=nx^{n-1}\)
\(\bullet\text{ Derivatives- Constant Multiple Rule}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{d}{dx}(cf(x))=cf'(x)\)
\(\bullet\text{ Derivatives- Sum and Difference Rules}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{d}{dx}[f(x) \pm g(x)]=f'(x) \pm g'(x)\)
\(\bullet\text{ Derivatives- Sin and Cos}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{d}{dx}sin(x)=cos(x)\)
\(\bullet\text{ Derivatives- Product Rule}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{d}{dx}[f(x) \cdot g(x)]=f(x) \cdot g'(x)+f'(x) \cdot g(x)\)
\(\bullet\text{ Derivatives- Quotient Rule}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{d}{dx}\left[\displaystyle\frac{f(x)}{g(x)}\right]=\displaystyle\frac{g(x) \cdot f'(x)-f(x) \cdot g'(x)}{[g(x)]^2}\)
\(\bullet\text{ Derivatives- Chain Rule}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{d}{dx}[f(g(x))]= f'(g(x)) \cdot g'(x)\)
\(\bullet\text{ Derivatives- ln(x)}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{d}{dx}[ln(x)]= \displaystyle \frac{1}{x}\)
\(\bullet\text{ Implicit Differentiation}\)
\(\,\,\,\,\,\,\,\,\)
\(\bullet\text{ Horizontal Tangent Line}\)
\(\,\,\,\,\,\,\,\,\)
\(\bullet\text{ Mean Value Theorem}\)
\(\,\,\,\,\,\,\,\,\)
\(\bullet\text{ Related Rates}\)
\(\,\,\,\,\,\,\,\,\)
\(\bullet\text{ Increasing and Decreasing Intervals}\)
\(\,\,\,\,\,\,\,\,\)
\(\bullet\text{ Intervals of concave up and down}\)
\(\,\,\,\,\,\,\,\,\)
\(\bullet\text{ Inflection Points}\)
\(\,\,\,\,\,\,\,\,\)
\(\bullet\text{ Graph of f(x), f'(x) and f”(x)}\)
\(\,\,\,\,\,\,\,\,\)
\(\bullet\text{ Newton’s Method}\)
\(\,\,\,\,\,\,\,\,x_{n+1}=x_n – \displaystyle \frac{f(x_n)}{f'(x_n)}\)
In Summary
Derivatives of exponential functions is a fundamental concept in calculus. The most common type of exponential in this section is \(f(x)=e^x\), but we also look at bases other than e like for example \(f(x)=2^x\).
There derivative of \(e^x\) is \(e^x\)
\(\frac{d}{dx}e^x=e^x\)
The derivative of an exponential with a based other than \(e\) is
\(\frac{d}{dx}b^x=b^x \cdot \ln(b)\)
For example,
\(\frac{d}{dx}2^x=2^x \cdot \ln(2)\)
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