Formulas and examples of the derivatives of exponential functions, in calculus, are presented. Several examples, with detailed solutions, involving products, sums, and quotients of exponential functions are examined.
Derivative of Exponential Functions to any Base
The derivative of \( f(x) = b^{x} \) is given by:
Note: if \( f(x) = e^{x} \), then \( f'(x) = e^{x} \).
Examples with Solutions
Example 1
Find the derivative of \( f(x) = 2^{x} \)
Solution to Example 1
Apply the formula above to obtain:
\[ f'(x) = 2^{x} \ln 2 \]Example 2
Find the derivative of \( f(x) = 3^{x} + 3x^{2} \)
Solution to Example 2
Let \( g(x) = 3^{x} \) and \( h(x) = 3x^{2} \). Function \( f \) is the sum of functions \( g \) and \( h \): \( f(x) = g(x) + h(x) \).
Use the sum rule, \( f'(x) = g'(x) + h'(x) \), to find the derivative of function \( f \):
\[ f'(x) = 3^{x} \ln 3 + 6x \]Example 3
Find the derivative of \( f(x) = \dfrac{e^{x}}{1 + x} \)
Solution to Example 3
Let \( g(x) = e^{x} \) and \( h(x) = 1 + x \). Function \( f \) is the quotient of functions \( g \) and \( h \): \( f(x) = \dfrac{g(x)}{h(x)} \). Hence we use the quotient rule:
\[ f'(x) = \dfrac{h(x)g'(x) - g(x)h'(x)}{[h(x)]^2} \]We find the individual derivatives:
\[ g'(x) = e^{x} \] \[ h'(x) = 1 \]Substitute into the quotient rule formula:
\[ f'(x) = \dfrac{(1 + x)(e^{x}) - (e^{x})(1)}{(1 + x)^{2}} \] \[ = \dfrac{xe^{x}}{(1 + x)^{2}} \]Example 4
Find the derivative of \( f(x) = e^{2x + 1} \)
Solution to Example 4
Let \( u = 2x + 1 \) and \( y = e^{u} \). Use the chain rule to find the derivative of function \( f \):
\[ f'(x) = \dfrac{dy}{du} \dfrac{du}{dx} \]Since \( \dfrac{dy}{du} = e^{u} \) and \( \dfrac{du}{dx} = 2 \):
\[ f'(x) = 2e^{2x + 1} \]Exercises
Find the derivative of each function:
- \( f(x) = e^{x} 2^{x} \)
- \( g(x) = 3^{x} - 3x^{3} \)
- \( h(x) = \dfrac{e^{x}}{2x - 3} \)
- \( j(x) = e^{(x^{2} + 2)} \)
Solutions to the Above Exercises
- \( f'(x) = e^{x} 2^{x} (\ln 2 + 1) \)
- \( g'(x) = 3^{x} \ln 3 - 9x^{2} \)
- \( h'(x) = \dfrac{e^{x}(2x - 5)}{(2x - 3)^{2}} \)
- \( j'(x) = 2x e^{(x^{2} + 2)} \)