111 LearnersLast updated on August 30, 2025
Derivative of 4e^x
We use the derivative of 4e^x, which is 4e^x, as a tool to measure how the function changes in response to a slight change in x. Derivatives help us calculate growth or decay in real-life situations. We will now talk about the derivative of 4e^x in detail.
What is the Derivative of 4e^x?
We now understand the derivative of 4e^x. It is commonly represented as d/dx (4e^x) or (4e^x)', and its value is 4e^x. The function 4e^x has a clearly defined derivative, indicating it is differentiable within its domain.
The key concepts are mentioned below:
Exponential Function: (e^x is the base of the natural logarithm).
Constant Multiple Rule: Rule for differentiating functions with a constant coefficient.
Derivative of e^x: The derivative of e^x is e^x.
Derivative of 4e^x Formula
The derivative of 4e^x can be denoted as d/dx (4e^x) or (4e^x)'.
The formula we use to differentiate 4e^x is: d/dx (4e^x) = 4e^x (or) (4e^x)' = 4e^x. This formula applies to all x.
Proofs of the Derivative of 4e^x
We can derive the derivative of 4e^x using proofs. To show this, we will use the fundamental rules of differentiation. There are several methods we use to prove this, such as:
Using the Constant Multiple Rule Using the Chain Rule We will now demonstrate that the differentiation of 4e^x results in 4e^x using the above-mentioned methods:
Using the Constant Multiple Rule The derivative of 4e^x can be easily derived using the Constant Multiple Rule. According to this rule, if you have a constant multiplied by a function, you can take the derivative of the function and multiply it by the constant.
To find the derivative of 4e^x, consider f(x) = 4e^x. The Constant Multiple Rule states: d/dx [c·f(x)] = c·f'(x) Here, c = 4 and f(x) = e^x, and we know that f'(x) = e^x.
Substituting in the rule gives: d/dx (4e^x) = 4·e^x = 4e^x.
Hence, proved. Using the Chain Rule To prove the differentiation of 4e^x using the Chain Rule, Consider the function as a composition: y = 4e^x = 4·u, where u = e^x
Differentiate y with respect to u, and then u with respect to x. dy/du = 4 (since 4 is a constant) du/dx = e^x (since the derivative of e^x is e^x)
Applying the chain rule: dy/dx = dy/du * du/dx = 4 * e^x = 4e^x.
Thus, we have shown using the Chain Rule that the derivative of 4e^x is 4e^x.
Higher-Order Derivatives of 4e^x
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 4e^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 4e^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 = 0, the derivative of 4e^x = 4e^0, which is 4. The derivative is always positive, indicating exponential growth for all x.
Common Mistakes and How to Avoid Them in Derivatives of 4e^x
Students frequently make mistakes when differentiating 4e^x. 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 Constant Multiple Rule
Students may forget to apply the constant multiple rule, which can lead to incorrect results. They often do not multiply the derivative of e^x by the constant 4. Ensure that each step is written in order, and the constant is correctly applied.
Mistake 2
Incorrect use of the Chain Rule
When differentiating composite functions involving 4e^x, students might misapply the chain rule.
For example: Incorrect differentiation: d/dx (4e^x) = 4xe^x. To avoid this mistake, ensure that you apply the chain rule correctly, recognizing the inner and outer functions.
Mistake 3
Ignoring the Exponential Function's Derivative
Some students fail to recall that the derivative of e^x is e^x.
This leads to errors in calculations.
Always remember that e^x differentiates to e^x, and apply this knowledge when differentiating 4e^x.
Mistake 4
Confusing Exponential Base with Variable
There is a common mistake where students confuse the base of the exponential function e with a variable.
For instance, they might incorrectly write d/dx (4e^x) as 4x·e^(x-1). Remember that e is a constant, and its derivative does not reduce the exponent by one.
Mistake 5
Overcomplicating the Differentiation Process
Students sometimes overcomplicate the differentiation of simple functions like 4e^x by unnecessarily applying complex rules. Focus on understanding the basic principles, such as the constant multiple rule, to keep the process straightforward.
Examples Using the Derivative of 4e^x
Problem 1
Calculate the derivative of (4e^x·x^2).
Here, we have f(x) = 4e^x·x².
Using the product rule, f'(x) = u′v + uv′.
In the given equation, u = 4e^x and v = x².
Let’s differentiate each term, u′ = d/dx (4e^x) = 4e^x v′ = d/dx (x²) = 2x
Substituting into the given equation, f'(x) = (4e^x)·(x²) + (4e^x)·(2x)
Let's simplify terms to get the final answer, f'(x) = 4e^x x² + 8xe^x
Thus, the derivative of the specified function is 4e^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 population of bacteria grows exponentially and is modeled by the function P(t) = 4e^t where t is time in hours. If t = 3 hours, find the rate of change of the population.
We have P(t) = 4e^t (growth of the population)...(1)
Now, we will differentiate the equation (1)
Take the derivative of P(t): dP/dt = 4e^t Given t = 3, substitute this into the derivative dP/dt = 4e^3
Thus, the rate of change of the population at t = 3 is 4e^3.
Explanation
We find the rate of change of the population at t = 3 by differentiating the function and substituting the given time value into the derivative.
This provides the exponential growth rate at that specific time.
Problem 3
Derive the second derivative of the function y = 4e^x.
The first step is to find the first derivative, dy/dx = 4e^x...(1)
Now we will differentiate equation (1) to get the second derivative: d²y/dx² = d/dx [4e^x] d²y/dx² = 4e^x
Therefore, the second derivative of the function y = 4e^x is 4e^x.
Explanation
We use the step-by-step process, where we start with the first derivative.
Given that the derivative of 4e^x is 4e^x, the second derivative follows the same pattern.
We then substitute the identity and simplify the terms to find the final answer.
Problem 4
Prove: d/dx (8e^x) = 8e^x.
Let’s start using the Constant Multiple Rule: Consider y = 8e^x
To differentiate, we recognize that the derivative of e^x is e^x, and apply the Constant Multiple Rule:
dy/dx = 8·d/dx [e^x]
Since the derivative of e^x is e^x, dy/dx = 8e^x
Hence proved.
Explanation
In this step-by-step process, we used the Constant Multiple Rule to differentiate the equation.
We then replace e^x with its derivative. As a final step, we apply the constant to derive the equation.
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 = (xe^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 4e^x
1.Find the derivative of 4e^x.
2.Can we use the derivative of 4e^x in real life?
3.Is it possible to take the derivative of 4e^x at any point?
4.What rule is used to differentiate e^x/x?
5.Are the derivatives of 4e^x and e^x the same?
6.Can we find the derivative of the 4e^x formula?
Important Glossaries for the Derivative of 4e^x
- Derivative: The derivative of a function indicates how the given function changes in response to a slight change in x.
- Exponential Function: A function in which an independent variable appears in the exponent.
- Constant Multiple Rule: A rule stating that the derivative of a constant times a function is the constant times the derivative of the function.
- Chain Rule: A rule for differentiating compositions of functions.
- Quotient Rule: A rule used to differentiate functions that are divided by each other.
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.
