Derivatives – Exponential Functions

Exponential and logarithmic derivatives are major derivative rules used throughout calculus. The derivative of \(e^x\) is itself, while other exponential bases use a factor of \(\ln(a)\). These problems also include logarithmic derivatives, chain rule, product rule, quotient rule, and power rule variations.

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

\(\textbf{1)}\) \(f(x)=\displaystyle2e^x\)

 

\(\textbf{2)}\) \(f(x)=\displaystyle3^x\)

 

\(\textbf{3)}\) \(f(x)=\displaystyle 2e^x+3^x\)

 

\(\textbf{4)}\) \(f(x)=\displaystyle\frac{1}{e^x}\)

 

\(\textbf{5)}\) \(f(x)=\displaystyle \frac{1}{3^x}\)

 

\(\textbf{6)}\) \(f(x)=\displaystyle e^x \ln x\)

 

\(\textbf{7)}\) \(f(x)=\displaystyle e^x + \ln x\)

 

\(\textbf{8)}\) \(f(x)=\displaystyle e^{3x}\)

 

\(\textbf{9)}\) \(f(x)=\displaystyle\pi^x\)

 

\(\textbf{10)}\) \(f(x)=\displaystyle x^{\pi}\)

 

\(\textbf{11)}\) \(f(x)=5e^{2x}\)

 

\(\textbf{12)}\) \(f(x)=4^x+7\ln x\)

 

\(\textbf{13)}\) \(f(x)=\ln(5x^2+1)\)

 

\(\textbf{14)}\) \(f(x)=e^x\cdot 2^x\)

 

\(\textbf{15)}\) \(f(x)=\displaystyle\frac{e^x}{x}\)

 

\(\textbf{16)}\) \(f(x)=\log_2 x\)

 

\(\textbf{17)}\) \(f(x)=\ln(e^x+x^2)\)

 

\(\textbf{18)}\) \(f(x)=e^{x^2}\ln x\)

 

\(\textbf{19)}\) \(f(x)=\displaystyle\frac{\ln x}{e^x}\)

 

\(\textbf{20)}\) \(f(x)=2^{x^2+1}\)

 

See Related Pages\(\)

\(\bullet\text{ Calculus Homepage}\)
\(\,\,\,\,\,\,\,\,\text{All the Best Topics…}\)
\(\bullet\text{ Definition of Derivative}\)
\(\,\,\,\,\,\,\,\, \displaystyle \lim_{\Delta x\to 0} \frac{f(x+ \Delta x)-f(x)}{\Delta x} \)
\(\bullet\text{ Equation of the Tangent Line}\)
\(\,\,\,\,\,\,\,\,f(x)=x^3+3x^2−x \text{ at the point } (2,18)\)
\(\bullet\text{ Derivatives- Constant Rule}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{d}{dx}(c)=0\)
\(\bullet\text{ Derivatives- Power Rule}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{d}{dx}(x^n)=nx^{n-1}\)
\(\bullet\text{ Derivatives- Constant Multiple Rule}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{d}{dx}(cf(x))=cf'(x)\)
\(\bullet\text{ Derivatives- Sum and Difference Rules}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{d}{dx}[f(x) \pm g(x)]=f'(x) \pm g'(x)\)
\(\bullet\text{ Derivatives- Sin and Cos}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{d}{dx}sin(x)=cos(x)\)
\(\bullet\text{ Derivatives- Product Rule}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{d}{dx}[f(x) \cdot g(x)]=f(x) \cdot g'(x)+f'(x) \cdot g(x)\)
\(\bullet\text{ Derivatives- Quotient Rule}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{d}{dx}\left[\displaystyle\frac{f(x)}{g(x)}\right]=\displaystyle\frac{g(x) \cdot f'(x)-f(x) \cdot g'(x)}{[g(x)]^2}\)
\(\bullet\text{ Derivatives- Chain Rule}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{d}{dx}[f(g(x))]= f'(g(x)) \cdot g'(x)\)
\(\bullet\text{ Derivatives- ln(x)}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{d}{dx}[ln(x)]= \displaystyle \frac{1}{x}\)
\(\bullet\text{ Implicit Differentiation}\)
\(\,\,\,\,\,\,\,\,\)
\(\bullet\text{ Horizontal Tangent Line}\)
\(\,\,\,\,\,\,\,\,\)
\(\bullet\text{ Mean Value Theorem}\)
\(\,\,\,\,\,\,\,\,\)
\(\bullet\text{ Related Rates}\)
\(\,\,\,\,\,\,\,\,\)
\(\bullet\text{ Increasing and Decreasing Intervals}\)
\(\,\,\,\,\,\,\,\,\)
\(\bullet\text{ Intervals of concave up and down}\)
\(\,\,\,\,\,\,\,\,\)
\(\bullet\text{ Inflection Points}\)
\(\,\,\,\,\,\,\,\,\)
\(\bullet\text{ Graph of f(x), f'(x) and f”(x)}\)
\(\,\,\,\,\,\,\,\,\)Thumbnail of Graph of First and Second Derivatives
\(\bullet\text{ Newton’s Method}\)
\(\,\,\,\,\,\,\,\,x_{n+1}=x_n – \displaystyle \frac{f(x_n)}{f'(x_n)}\)

 

Scroll to Top