# Derivatives of Exponential Functions

Exponential functions have the form $$f\left( x \right) = {a^x},$$ where $$a$$ is the base. The base is always a positive number not equal to $$1.$$

If the base is equal to the number $$e:$$

$a = e \approx 2.718281828 \ldots ,$

then the derivative is given by

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

(This formula is proved on the page Definition of the Derivative.)

The function $$y = {e^x}$$ is often referred to as simply the exponential function.

Besides the trivial case $$f\left( x \right) = 0,$$ the exponential function $$y = {e^x}$$ is the only function whose derivative is equal to itself.

Now we consider the exponential function $$y = {a^x}$$ with arbitrary base $$a$$ $$\left( {a \gt 0, a \ne 1} \right)$$ and find an expression for its derivative.

As $$a = {e^{\ln a}},$$ then

$a^x = \left( {{e^{\ln a}}} \right)^x = e^{x\ln a}.$

Using the chain rule, we have

$\left( {{a^x}} \right)^\prime = \left( {{e^{x\ln a}}} \right)^\prime = {e^{x\ln a}} \cdot \left( {x\ln a} \right)^\prime = {e^{x\ln a}} \cdot \ln a = {a^x} \cdot \ln a.$

Thus

$\left( {{a^x}} \right)^\prime = {a^x}\ln a.$

In the examples below, determine the derivative of the given function.

## Solved Problems

Click or tap a problem to see the solution.

### Example 1

$y = {\pi ^{\frac{1}{x}}}$

### Example 2

$y = \sqrt {{2^x}}$

### Example 3

$y = {2^{\sqrt x }}\left( {x \gt 0} \right)$

### Example 4

$y = {e^{ - {x^3}}}$

### Example 5

$y = {4^{3{x^2}}}$

### Example 6

$y = 3^{\frac{1}{x}}$

### Example 1.

$y = {\pi ^{\frac{1}{x}}}$

Solution.

By the chain rule, we obtain:

$y'\left( x \right) = \left( {{\pi ^{\frac{1}{x}}}} \right)^\prime = {\pi ^{\frac{1}{x}}} \cdot \ln \pi \cdot {\left( {\frac{1}{x}} \right)^\prime } = {\pi ^{\frac{1}{x}}} \cdot \ln \pi \cdot \left( { - \frac{1}{{{x^2}}}} \right) = - \frac{{{\pi ^{\frac{1}{x}}}\ln \pi }}{{{x^2}}}.$

### Example 2.

$y = \sqrt {{2^x}}$

Solution.

$y'\left( x \right) = \left( {\sqrt {{2^x}} } \right)^\prime = \frac{1}{{2\sqrt {{2^x}} }} \cdot {\left( {{2^x}} \right)^\prime } = \frac{1}{{2\sqrt {{2^x}} }} \cdot {2^x}\ln 2 = \frac{{{2^x}\ln 2}}{{2\sqrt {{2^x}} }} = \frac{{\sqrt {{2^x}} \ln 2}}{2}.$

### Example 3.

$y = {2^{\sqrt x }}\left( {x \gt 0} \right)$

Solution.

Using the chain rule, we get

$y^\prime = \left( {{2^{\sqrt x }}} \right)^\prime = {{2^{\sqrt x }}\ln 2 \cdot \left( {\sqrt x } \right)^\prime } = {2^{\sqrt x }}\ln 2 \cdot \frac{1}{{2\sqrt x }} = \frac{{{2^{\sqrt x }}\ln 2}}{{2\sqrt x }} = \frac{{{2^{\sqrt x - 1}}\ln 2}}{{\sqrt x }}.$

### Example 4.

$y = {e^{ - {x^3}}}$

Solution.

By the chain rule,

$y^\prime = \left( {{e^{ - {x^3}}}} \right)^\prime = {e^{ - {x^3}}} \cdot \left( { - {x^3}} \right)^\prime = {e^{ - {x^3}}} \cdot \left( { - 3{x^2}} \right) = - 3{x^2}{e^{ - {x^3}}}.$

### Example 5.

$y = {4^{3{x^2}}}$

Solution.

Using the chain rule, we have

$y^\prime = \left( {{4^{3{x^2}}}} \right)^\prime = {4^{3{x^2}}}\ln 4 \cdot \left( {3{x^2}} \right)^\prime = {4^{3{x^2}}}\ln 4 \cdot 6x = 6x{4^{3{x^2}}}\ln 4.$

### Example 6.

$y = 3^{\frac{1}{x}}$

Solution.

By the chain rule,

$y^\prime = \left( {{3^{\frac{1}{x}}}} \right)^\prime = {3^{\frac{1}{x}}}\ln 3 \cdot \left( {\frac{1}{x}} \right)^\prime = {3^{\frac{1}{x}}}\ln 3 \cdot \left( { - \frac{1}{{{x^2}}}} \right) = - \frac{{{3^{\frac{1}{x}}}\ln 3}}{{{x^2}}}.$