# Derivatives of Exponential Functions

**Definition:** An exponential function has the form $f(x) = ab^x$.
Notice the difference between power functions and exponential functions.

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## **Derivative Rule for Exponential Functions**: If $f(x) = ab^x$, then $f'(x) = a\ln(b)b^x$.

Another way of saying this is: $\displaystyle\frac{d}{dx} \left( ab^x \right) = a \ln(b) b^x$.

**Q1:** If $f(x) = 4^x$, what is $f'(x)$?
**A1:** $f'(x) = \ln(4) 4^x$.

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**Q2:** Find $\displaystyle\frac{d}{dx} \left( 2e^x \right)$.
**A2:** The answer is $2 \ln(e) e^x$. Since $\ln(e) = 1$, we would usually just write this as $2e^x$.

## :warning: Be careful about what is actually being raised to the exponent. A common mistake is to write this as $\frac{d}{dx} \left( 2e^x \right) = \ln(2e) 2e^x$. But only the "e" is being raised to the x; the "2" is not.

## Exponential Functions That Don't Fit the Form


Sometimes a function will be exponential, but it won't be immediately obvious.

## **Example 1:** What is $\displaystyle\frac{d}{dx} \left( 2^{-x} \right)$ ?

**Solution:** Using algebra, we can rewrite $2^{-x}$ as $\left( 2^{-1} \right)^x$. Now this fits the exponential form with $b = 2^{-1}$. So taking the derivative gives us $\ln(2^{-1}) \left( 2^{-1} \right)^x$.

**Answer:** $\displaystyle\frac{d}{dx} \left( 2^{-x} \right) = \ln(2^{-1}) (2^{-1})^x$

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## **Example 2:** Find the derivative of $4^x 3^x$.

**Solution:** We can rewrite $4^x 3^x$ as $12^x$. Then we take the derivative using exponential rule: $\displaystyle\frac{d}{dx} \left( 12^x \right) = \ln(12) 12^x$.

**Answer:** The derivative of $4^x 3^x$ is $\ln(12) 12^x$.

## Example 3: Find the derivative of $\displaystyle\frac{10^x}{9^x}$.

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## Finding the Equation of the Tangent Line to an Exponential Curve

One of the major applications of derivative is finding the equation of the tangent line. Here are some examples of how to do that for exponential curves.

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For any curve, the basic process to finding the equation of the tangent line to a curve at a point is:
1. Find the point by plugging the x-value into the function.
2. Find the slope by plugging the x-value into the derivative.
3. Use [point-slope form](https://www.mathsisfun.com/algebra/line-equation-point-slope.html) to find the equation of the tangent line.