An exponential function is defined as:
$f(x)=b^x$
where b is called the base and x can be any real number. The base b should be greater than 0 and not equal to 1.
Graphs
Graph of $f(x)=2^x$
Graph of $f(x)=3^x$
Graph of $f(x)=10^x$
Comparison Graph $f(x)=2^x$,$f(x)=3^x$,$f(x)=10^x$
Keypoints
(1)Domain of the exponential function is R, the set of all real numbers.
(2) Range of the exponential function is the set of all positive real numbers.
(3) The point (0, 1) is always on the graph of the exponential function (this is arestatement of the fact that $b^0= 1$ for any real b > 0).
(4) if b > 1 Exponential function is ever increasing; i.e., as we move from left to right, the graph rises above.
(5) For very large negative values of x, the exponential function is very close to 0. In other words, in the second quadrant, the graph approaches x-axis (but never meets it)
(6) if 0
Common Exponential Function
An common exponential function is defined as:
$f(x)=10^x$
Natural Exponential function
A Natural exponential function is defined as:
$f(x)=e^x$
Where where e is Euler's number, approximately equal to 2.71828
Derivative of Natural exponential function
$\frac {d}{dx} e^x = e^x$
Derivative of exponential function
$\frac {d}{dx} a^x = a^x ln a$
Solved Examples
Question 1
Find the derivative of \( f(x) = e^{2x} \). Solution
\[ \frac{d}{dx}e^{2x} = e^{2x} \cdot \frac{d}{dx}(2x) = 2e^{2x} \]
Question 2
Find the derivative of \( g(x) = x \cdot e^x \). Solution
Here, you need to apply the product rule, which is \( (uv)' = u'v + uv' \), where \( u = x \) and \( v = e^x \).
\[ \frac{dg}{dx} = 1 \cdot e^x + x \cdot e^x = e^x + xe^x \]
Question 3
Find the derivative of \( h(x) = 2^x \). Solution
\[ \frac{dh}{dx} = 2^x \ln(2) \]
Question 4
Find the derivative of \( k(x) = \frac{e^x}{x} \) for \( x \neq 0 \). Solution
Apply the quotient rule, \( \left( \frac{u}{v} \right)' = \frac{u'v - uv'}{v^2} \), where \( u = e^x \) and \( v = x \).
\[ \frac{dk}{dx} = \frac{e^x \cdot x - e^x \cdot 1}{x^2} = \frac{xe^x - e^x}{x^2} = \frac{e^x(x - 1)}{x^2} \]
