The derivative of a function is one of the fundamental mathematical concepts. The process by which we find the derivative of a given function is known as differentiation. You may have heard about integration, which is the inverse of differentiation.  Finding a derivative of the function means to calculate the rate of change of the function at any given point. We calculate the rate of change by dividing the change in the independent variable represented as \triangle y by the change in the dependent variable \triangle x.

m = \frac {\triangle y}{\triangle x}

The derivative function is represented by \frac{d(y)}{d(x)} f(x) and is also known as instantaneous rate of change. The instantaneous rate of change is the rate of change in a function at a specific point.

What are Exponential Functions

In this article, we will learn how to find the derivative of exponential functions. But before proceeding to how to find the derivative, let us recall what exponential functions are. A function in which the independent variable x is an exponent is known as an exponential function. The general notation of exponential function is given below:

y = 2 ^ x

Here, y is the dependent variable, x is an independent variable and 2 is the base of the function.

Before finding the derivative of exponential functions, you should know how to find the derivative of the linear functions. It is because finding the derivative of exponential function also involves finding the derivative of its exponent which is often in the linear form. The slope intercept form of the linear functions is:

y = mx + b

Here, m is the slope of a line, x is the independent variable and b is the constant.  The derivative of a linear equation is equal to its slope m and the derivative of a constant function is 0.

Method of Finding Derivative of Exponential Function

To find the derivative of an exponential function, follow these rules step by step:

Step 1

First, write the exponential function as it is. For example, if we need to find the derivative of the exponential function e ^ x, then we need to write this function as it is.

Step 2

Calculate the logarithm of the base of the exponential function. For example, in the exponential function e ^ x, the base is e. Hence, the log of the base will be written as ln e

Step 3

Calculate the derivative of the exponent. In the exponential function, e ^x, the exponent is x. The derivative of x will be 1 because it is of the form mx. Since, the value of m is 1, hence the derivative of x is 1.

Step 4

Multiply the expressions obtained in step 1, step 2 and step 3 together to get the derivative of an exponential function. In the example, e ^x, the derivative will be:

= e ^ x \cdot ln e \cdot 1

Here, e is a natural number whose value is equal to 2.71828.. The natural logarithms are represented as ln (e) and their values are equal to 1. Hence, the derivative of e ^x = e ^ x.

Derivatives of Common Functions

We will go through some of the examples in this article which will clarify the concept of derivatives of exponential functions further. Before proceeding to examples, let us look at the following table which describes the derivatives of some common functions.

   
Names of functionsNotationDerivatives
Constantb0
Linear functionmx + bm
Power
Trigonometry (i)sin (x)cos (x)
Trigonometry (ii)cos (x)- sin (x)
Trigonometry (iii)tan (x)
Square root
Logarithmsln(x)
Logarithms

Derivatives Rules

Derivative rules are also known as differentiation rules. The following table shows some of the rules of derivatives:

RulesFunctionsDerivatives
Multiplied by constantaf
Power rule
Sum rulea + b
Difference rulea - b
Product ruleab
Quotient rulea/b
Reciprocal rule1/a

You can see that the derivative of a function whose power is 3 is a quadratic function.

Example 1

Find the derivative of the following function:

e ^ {2x + 1}

Solution

Follow the following step by step solution to find the derivative of the above function:

Step 1 - To find the derivative, first write the function e ^ {2x + 1} as it is.

Step 2 - Now, find the derivative of the function in the power. The function in the power is a linear function 2x + 1. Since, the derivative of a linear function is equal to its slope, hence the derivative of 2x + 1 will be 2.

Step 3 - Take the log of the base. In the above example, the base is e which is a natural number. The natural log denoted by ln (e) is equal to 1.

Step 4 - Multiply the expressions obtained in step 1, 2 and 3 together to get the derivative of the exponential function:

= e ^ {2x + 1} \cdot 2 \cdot 1

\frac {d}{dx} =  2 e ^ {2x + 1}

Example 2

Find the derivative of the following function:

6 ^ {x ^ 3}

Solution

Follow these steps to find how to differentiate the above function:

Step 1 - To find the derivative, first write the function 6 ^ {x ^ 3} as it is.

Step 2 - Find the log of the base. In the above exponential function, the base is 6, hence in the logarithmic form, it will be written as ln (6).

Step 3 - Find the derivative of the function in power. The power of the function is x ^ 3. We know that the derivative of the power function is denoted by x ^ n = n x ^ {n - 1}. Hence, using this rule we will get the derivative of the function x ^ 3 = 3 x ^ 2.

Step 4 - Multiply the expressions obtained in step 1, 2 and 3 together to get the derivative of the exponential function:

\frac {d}{dx} 6 ^ {x} ^ 3 = 6 ^ {x} ^ 3 \cdot 3 x ^ 2 \cdot ln (6)

In the simplified form, we can write the answer as:

= 3 ln (6) x ^ 2 \cdot 6 ^ {x ^3}

 

Example 3

Find the derivative of the following function:

2 ^ {3x + 7}

Solution

Step 1 - To find the derivative, first write the function 2 ^ {3x + 7} as it is.

Step 2 - Now, find the derivative of the function in the power. The function in the power is a linear function 3x + 7. Since, the derivative of a linear function is equal to its slope, hence the derivative of 3x + 7 will be 3.

Step 3 - Take the log of the base. In the above example, the base is 2. The log of 2 will be written as ln (2).

Step 4 - Multiply the expressions obtained in step 1, 2 and 3 together to get the derivative of the exponential function:

\frac {d}{d(x)} = 2 ^ {3x + 7} \cdot 3 \cdot ln (2)

\frac {d}{dx} =   2 ^ {3x + 1} \cdot 3 \cdot ln(2)

 

Example 4

Find the derivative of the following function:

4 ^ {3x - x ^ 2}

Solution

Follow these steps to find the derivative of the above function. We will use the rules of derivatives throughout this problem.

Step 1 - First, write the function as it is 4 ^ {3x - x ^ 2}.

Step 2 - Next, we need to find the derivative of the function in power. The function in the power is 3x - x ^ 2. According to the differential rule of derivative, we need to find the derivatives of both the functions separately. The derivative of the function 3x is 3 and the derivative of the function x ^ 2 = 2x. Hence, we can write the expression in this step as 3 - 2x.

Step 3 - Find the log of the base. The log of 4 can be denoted as ln (4).

Step 4 - Multiply the expressions from step 1, 2 and 3 together to get the derivative of entire exponential function:

\frac {d} {d (x)} = 4 ^ {3x - x ^ 2} \cdot 3 - 2x \cdot ln (4)

 

Example 5

Find the derivative of the following function:

4 ^ {sin x}

Solution

We will apply the differential trigonometry rule to solve this example.

Step 1 - First, write the function as it is 4 ^ {sin x}.

Step 2 - Now, find the derivative of the function in the exponent. The exponent of the function is a trigonometric function  sin x. According to the derivative rules of trigonometric functions, the derivative of a sine function is equal to the cosine of the same function. Hence, \frac {d}{d(x)} sin x = cos x.

Step 3 - In this step, we will find the log of the base. Since in the above exponential function, the base is 4, hence when we will take the log, we will write it as ln (4).

Step 4 - Multiply the expressions obtained in step 1, 2 and 3 together to get the derivative of the whole exponential function:

\frac{d}{d(x)} = 4 ^ {sin x} \cdot cos x \cdot ln (4)

Example 6

Find the derivative of the following function:

x ^ 4 3 ^ {5x}

Solution

The above function is a product of two exponential functions. Therefore, we will use the derivative product function rule to find the derivative of the above function. According to the derivative product rule:

\frac {d}{dx} (a \cdot b) = a' b + ab'

Step 1 - First find the derivative of x ^ 4. We will use the derivative power rule to find the derivative of x ^ 4. According to the derivative power functions rule, x ^ n = n x ^ {x - 1}. Hence, the derivative of x ^ 4 = 4 x ^ 3.

Step 2 - In this step, we need to find the derivative of 3 ^ {5x}. The derivative of the exponent 5x is equal to 5. The log of base 3 will be written as ln (3). The original function will be multiplied as it is because it is an exponential function. Hence, the derivative of the entire function is 5 \cdot ln (3) \cdot 3 ^ {5x}.

Step 3 - Use the product rule \frac {d}{dx} (a \cdot b) = a' b + ab' to write the derivative of the function like this:

\frac {d}{d(x)}= 4 \cdot x ^ 3 \cdot 3 ^ {5x} + x ^ 4 \cdot 5 \cdot ln (3) \cdot 3 ^ {5x}

 

Example 7

Find the derivative of the following function:

\frac { 2 ^ x } { 3 ^ {-x}}

Solution

Since the above function is in the fractional form, hence we will apply the derivative quotient rule for differentiating this function. According to the derivative quotient rule:

\frac {a} {b} = \frac {a b ' - a' b} { b ^ 2}

In the above example, a = 2 ^ x and b = 3 ^ {-x}.

Step 1 - Find the derivative of the numerator a like this by applying the derivative sum rule:

a ' = 2 ^ x \cdot 1 \cdot ln (2)

Step 2 - Find the derivative of the denominator b like this:

b' = 3 ^ {-x} \cdot -1 \cdot ln (3)

Step 3 - Use the derivative quotient rule \frac {a} {b} = \frac {a' b  - a b'} { b ^ 2} to write the derivative of the entire exponential function:

= \frac { 3 ^ {-x} \cdot 2 ^ x ln (2) + 3 ^ {-x} \cdot 2 ^ x \cdot ln (3)} { 3 ^ {-2x}}

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Emma

I am passionate about travelling and currently live and work in Paris. I like to spend my time reading, gardening, running, learning languages and exploring new places.

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