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106 LearnersLast updated on 17 September 2025
Derivative of Exponents
We use the derivative of exponential functions, such as e^x, as a measuring tool for how exponential functions change in response to a slight change in x. Derivatives help us calculate profit or loss in real-life situations. We will now discuss the derivative of exponential functions in detail.
What is the Derivative of Exponential Functions?
We now understand the derivative of exponential functions. The derivative of e^x is commonly represented as d/dx (\(e^x\)) or (\(e^x\))', and its value is \(e^x\).
Exponential functions have clearly defined derivatives, indicating they are differentiable across their domain.
The key concepts are mentioned below:
Exponential Function: An exponential function such as \(e^x\).
Chain Rule: Rule for differentiating complex exponential functions.
Natural Logarithm: The inverse function of the exponential function ln(x).
Derivative of Exponential Function Formula
The derivative of e^x can be denoted as d/dx (\(e^x\)) or (\(e^x\))'. The formula we use to differentiate e^x is: d/dx (\(e^x\)) = \(e^x \)The formula applies to all x.
Proofs of the Derivative of Exponential Functions
We can derive the derivative of \(e^x\) using proofs.
To show this, we will use the properties of exponential functions along with the rules of differentiation.
There are several methods we use to prove this, such as:
By First Principle
Using Chain Rule
Using Product Rule
We will now demonstrate that the differentiation of \(e^x\) results in e^x using the above-mentioned methods:
By First Principle
The derivative of \(e^x\) can be proved using the First Principle, which expresses the derivative as the limit of the difference quotient.
To find the derivative of \(e^x\) using the first principle, we will consider f(x) = \(e^x\).
Its derivative can be expressed as the following limit. f'(x) = limₕ→₀ [f(x + h) - f(x)] / h … (1)
Given that f(x) = \(e^x\), we write f(x + h) = \(e^(x + h)\).
Substituting these into equation (1), f'(x) = limₕ→₀ [\(e^(x + h) - e^x\)] / h = limₕ→₀ [\(e^x · (e^h - 1)\)] / h = \(e^x \)· limₕ→₀ [\((e^h - 1)\) / h]
Using the limit property, limₕ→₀ (\(e^h - 1\)) / h = 1. f'(x) = \(e^x\) · 1 = \(e^x\)
Hence, proved.
Using Chain Rule
To prove the differentiation of \(e^x \)using the chain rule,
We use the formula: If y = \(e^(u)\) where u = x, then dy/dx = dy/du · du/dx = \(e^u\) · 1 = \(e^x\)
Using Product Rule
We will now prove the derivative of \(e^x\) using the product rule.
The step-by-step process is demonstrated below:
Here, we consider the function as a product of two identical exponential functions: \(e^x · e^0 = e^x · 1\)
Using the product rule formula: d/dx [u·v] = u'·v + u·v' u = \(e^x\) and v = \(e^0\) u' = d/dx (\(e^x\)) = \(e^x \)v' = d/dx (\(e^0\)) = 0 d/dx (\(e^x\)) = \(e^x · 1 + e^x · 0 = e^x\)
Thus, d/dx (\(e^x\)) =\( e^x\).
Higher-Order Derivatives of Exponential Functions
When a function is differentiated several times, the derivatives obtained are referred to as higher-order derivatives. Higher-order derivatives can be a little tricky.
To understand them better, think of a car where the speed changes (first derivative) and the rate at which the speed changes (second derivative) also changes.
Higher-order derivatives make it easier to understand functions like \(e^x\).
For the first derivative of a function, we write f′(x), which indicates how the function changes or its slope at a certain point.
The second derivative is derived from the first derivative, which is denoted using f′′(x). Similarly, the third derivative, f′′′(x), is the result of the second derivative, and this pattern continues.
For the nth Derivative of e^x, we generally use fⁿ(x) for the nth derivative of a function f(x), which tells us the change in the rate of change (continuing for higher-order derivatives).
Special Cases:
When x is infinity, the derivative remains \(e^x\) because exponential functions grow indefinitely. When x is 0, the derivative of \(e^x = e^0\), which is 1.
Common Mistakes and How to Avoid Them in Derivatives of Exponential Functions
Students frequently make mistakes when differentiating exponential functions. These mistakes can be resolved by understanding the proper solutions. Here are a few common mistakes and ways to solve them:
Mistake 1
Not applying the Chain Rule
Students may forget to apply the chain rule when differentiating more complex exponential functions.
For example, d/dx (\(e^(3x)\)) should be \(3e^(3x)\), not simply \(e^(3x)\). Always identify the inner function and differentiate it along with the outer function.
Mistake 2
Mixing up Exponential and Logarithmic Derivatives
Students often confuse the derivatives of exponential functions with logarithmic functions. Remember that the derivative of \(e^x \)is \(e^x,\) while the derivative of ln(x) is 1/x. Keep these distinctions clear to avoid confusion.
Mistake 3
Ignoring the Constant Multiplier
In cases like differentiating\( 5e^x\), students might forget to multiply by 5 and incorrectly write the derivative as \(e^x\) instead of 5\(e^x\). Always check for constants and make sure they are included in the derivative.
Mistake 4
Incorrect Application of Product Rule
While differentiating products like \(e^x\) · x, students may misapply the product rule. The correct differentiation should use u = \(e^x\) and v = x to yield \(e^x · x + e^x · 1\). Ensure the product rule is applied accurately.
Mistake 5
Forgetting to Simplify
Students sometimes forget to simplify the expression after differentiation, leading to cumbersome results. Simplifying the expression can make further computation easier and reduce errors.
Examples Using the Derivative of Exponential Functions
Problem 1
Calculate the derivative of (e^x · x²).
Here, we have f(x) = \(e^x · x²\).
Using the product rule, f'(x) = u′v + uv′
In the given equation, u = \(e^x \)and v = x².
Let’s differentiate each term, \(u′ = d/dx (e^x) = e^x v′ = d/dx (x²) = 2x\)
Substituting into the given equation, \(f'(x) = (e^x) · (x²) + (e^x) · (2x)\)
Let’s simplify terms to get the final answer, \(f'(x) = e^x · x² + 2x · e^x\)
Thus, the derivative of the specified function is e^x · (x² + 2x).
Explanation
We find the derivative of the given function by dividing the function into two parts. The first step is finding its derivative and then combining them using the product rule to get the final result.
Problem 2
A company tracks its revenue using the function R(x) = e^(0.05x), where x represents time in months. Calculate the rate of change of revenue at x = 10 months.
We have R(x) = \(e^(0.05x)\) (rate of revenue change)...(1)
Now, we will differentiate the equation (1)
Take the derivative of \(e^(0.05x): dR/dx = 0.05e^(0.05x)\)
Given x = 10, substitute this into the derivative: \(dR/dx = 0.05e^(0.05*10) = 0.05e^0.5\)
Hence, the rate of change of revenue at 10 months is\( 0.05e^0.5\).
Explanation
We find the rate of change of revenue at x = 10 by differentiating the revenue function and substituting the given value of x into the derivative.
Problem 3
Derive the second derivative of the function y = e^(2x).
The first step is to find the first derivative, \(dy/dx = 2e^(2x)\)...(1)
Now we will differentiate equation (1) to get the second derivative:
\(d²y/dx² = d/dx [2e^(2x)] = 2 · 2e^(2x) = 4e^(2x)\)
Therefore, the second derivative of the function y = \(e^(2x) is 4e^(2x)\).
Explanation
We use the step-by-step process, where we start with the first derivative. Using basic differentiation rules, we differentiate \(2e^(2x)\) to find the second derivative.
Problem 4
Prove: d/dx (e^(3x)) = 3e^(3x).
Let’s start using the chain rule: Consider\( y = e^(3x)\)
To differentiate, we apply the chain rule:\( dy/dx = e^(3x) · d/dx (3x) = e^(3x) · 3 = 3e^(3x)\)
Hence proved.
Explanation
In this step-by-step process, we used the chain rule to differentiate the equation by multiplying the outer function's derivative by the inner function's derivative.
Problem 5
Solve: d/dx (e^x/x).
To differentiate the function, we use the quotient rule: \(d/dx (e^x/x) = (d/dx (e^x) · x - e^x · d/dx(x))/ x²\)
We will substitute \(d/dx (e^x) = e^x\) and \(d/dx (x) = 1 = (e^x · x - e^x · 1) / x² = (x · e^x - e^x) / x² = e^x(x - 1) / x²\)
Therefore, \(d/dx (e^x/x) = e^x(x - 1)/ x²\)
Explanation
In this process, we differentiate the given function using the quotient rule. As a final step, we simplify the equation to obtain the final result.
FAQs on the Derivative of Exponents
1.Find the derivative of e^x.
2.Can we use the derivative of exponents in real life?
3.Is it possible to take the derivative of e^x at infinity?
4.What rule is used to differentiate e^x/x?
5.Are the derivatives of e^x and ln(x) the same?
Important Glossaries for the Derivative of Exponents
- Derivative: The derivative of a function indicates how the given function changes in response to a slight change in x.
- Exponential Function: A function of the form \(e^x\), where e is the base of the natural logarithm.
- Chain Rule: A rule for differentiating compositions of functions, important for exponential functions with inner terms.
- Product Rule: A rule used for differentiating products of two functions.
- Quotient Rule: A rule used for differentiating the division of two functions.
Jaskaran Singh Saluja
About the Author
Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.
Fun Fact
: He loves to play the quiz with kids through algebra to make kids love it.
