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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

$\frac{d}{dx}(\ln x)={\displaystyle \frac{1}{x}}$

$\frac{d}{dx}({\log }_{a}x)=\frac{1}{x\ln a}$

Practice Problems

Find the derivative

$\mathbf{\text{1)}}$ $f(x)={\displaystyle 2e^x}$

Show Derivative

The answer is $f^{\prime }(x)={\displaystyle 2e^x}$

$\mathbf{\text{2)}}$ $f(x)={\displaystyle 3^x}$

Show Derivative

The answer is $f^{\prime }(x)={\displaystyle 3^x\cdot (\ln 3)}$

$\mathbf{\text{3)}}$ $f(x)={\displaystyle 2e^x+3^x}$

Show Derivative

The answer is $f^{\prime }(x)={\displaystyle 2e^x+3^x\cdot (\ln 3)}$

$\mathbf{\text{4)}}$ $f(x)={\displaystyle \frac{1}{e^x}}$

Show Derivative

The answer is $f^{\prime }(x)=-{\displaystyle \frac{1}{e^x}}$

$\mathbf{\text{5)}}$ $f(x)={\displaystyle \frac{1}{3^x}}$

Show Derivative

The answer is $f^{\prime }(x)=-{\displaystyle \frac{\ln 3}{3^x}}$

$\mathbf{\text{6)}}$ $f(x)={\displaystyle e^x\ln x}$

Show Derivative

The answer is $f^{\prime }(x)={\displaystyle e^x\ln x+\frac{e^x}{x}}$

Show Work

$\text{Use Product Rule}$

$\frac{d}{dx}(f(x)\cdot g(x))=f^{\prime }(x)\cdot g(x)+f(x)\cdot g^{\prime }(x)$

$f(x)=e^x,g(x)=\ln x$

$\frac{d}{dx}(e^x\cdot \ln x)={\left(e^x\right)}^{\prime }\cdot (\ln x)+\left(e^x\right)\cdot {(\ln x)}^{\prime }$

$\frac{d}{dx}(e^x\cdot \ln x)=\left(e^x\right)\cdot (\ln x)+\left(e^x\right)\cdot \left(\frac{1}{x}\right)$

The answer is $f^{\prime }(x)={\displaystyle e^x\ln x+\frac{e^x}{x}}$

$\mathbf{\text{7)}}$ $f(x)={\displaystyle e^x+\ln x}$

Show Derivative

The answer is $f^{\prime }(x)={\displaystyle e^x+\frac{1}{x}}$

$\mathbf{\text{8)}}$ $f(x)={\displaystyle e^{3x}}$

Show Derivative

The answer is $f^{\prime }(x)={\displaystyle 3e^{3x}}$

$\mathbf{\text{9)}}$ $f(x)={\displaystyle {\pi }^x}$

Show Derivative

The answer is $f^{\prime }(x)={\displaystyle {\pi }^x\cdot (\ln \pi )}$

$\mathbf{\text{10)}}$ $f(x)={\displaystyle x^{\pi }}$

Show Derivative

The answer is $f^{\prime }(x)={\displaystyle \pi x^{\pi -1}}$

See Related Pages$$

$\bullet \text{ Calculus Homepage}$
$\text{All the Best Topics…}$

$\bullet \text{ Definition of Derivative}$
${\displaystyle \underset{\mathrm{\Delta }x\to 0}{lim}\frac{f(x+\mathrm{\Delta }x)-f(x)}{\mathrm{\Delta }x}}$

$\bullet \text{ Equation of the Tangent Line}$
$f(x)=x^3+3x^2-x\text{ at the point }(2,18)$

$\bullet \text{ Derivatives- Constant Rule}$
${\displaystyle \frac{d}{dx}(c)=0}$

$\bullet \text{ Derivatives- Power Rule}$
${\displaystyle \frac{d}{dx}(x^n)=n{x}^{n-1}}$

$\bullet \text{ Derivatives- Constant Multiple Rule}$
${\displaystyle \frac{d}{dx}(cf(x))=c{f}^{\prime }(x)}$

$\bullet \text{ Derivatives- Sum and Difference Rules}$
${\displaystyle \frac{d}{dx}[f(x)\pm g(x)]=f^{\prime }(x)\pm g^{\prime }(x)}$

$\bullet \text{ Derivatives- Sin and Cos}$
${\displaystyle \frac{d}{dx}sin(x)=cos(x)}$

$\bullet \text{ Derivatives- Product Rule}$
${\displaystyle \frac{d}{dx}[f(x)\cdot g(x)]=f(x)\cdot g^{\prime }(x)+f^{\prime }(x)\cdot g(x)}$

$\bullet \text{ Derivatives- Quotient Rule}$
${\displaystyle \frac{d}{dx}\left[{\displaystyle \frac{f(x)}{g(x)}}\right]={\displaystyle \frac{g(x)\cdot f^{\prime }(x)-f(x)\cdot g^{\prime }(x)}{[g(x){]}^2}}}$

$\bullet \text{ Derivatives- Chain Rule}$
${\displaystyle \frac{d}{dx}[f(g(x))]=f^{\prime }(g(x))\cdot g^{\prime }(x)}$

$\bullet \text{ Derivatives- ln(x)}$
${\displaystyle \frac{d}{dx}[ln(x)]={\displaystyle \frac{1}{x}}}$

$\bullet \text{ Implicit Differentiation}$
$$

$\bullet \text{ Horizontal Tangent Line}$
$$

$\bullet \text{ Mean Value Theorem}$
$$

$\bullet \text{ Related Rates}$
$$

$\bullet \text{ Increasing and Decreasing Intervals}$
$$

$\bullet \text{ Intervals of concave up and down}$
$$

$\bullet \text{ Inflection Points}$
$$

$\bullet \text{ Graph of f(x), f'(x) and f”(x)}$
$$

$\bullet \text{ Newton’s Method}$
$x_{n+1}=x_n–{\displaystyle \frac{f(x_n)}{f^{\prime }(x_n)}}$

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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