102 LearnersLast updated on July 16th, 2025
Derivative of e^7x
We use the derivative of e^7x, which is 7e^7x, as a measuring tool for how the exponential function changes in response to a slight change in x. Derivatives help us calculate profit or loss in real-life situations. We will now talk about the derivative of e^7x in detail.
What is the Derivative of e^7x?
We now understand the derivative of e^7x. It is commonly represented as d/dx (e^7x) or (e^7x)', and its value is 7e^7x. The function e^7x 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 for natural logarithms.
Chain Rule: A fundamental rule for differentiating composite functions like e^7x.
Natural Exponential Function: e^x is the natural exponential function where the base is Euler's number (approximately 2.718).
Derivative of e^7x Formula
The derivative of e^7x can be denoted as d/dx (e^7x) or (e^7x)'.
The formula we use to differentiate e^7x is: d/dx (e^7x) = 7e^7x (or) (e^7x)' = 7e^7x
The formula applies to all x in the real number domain.
Proofs of the Derivative of e^7x
We can derive the derivative of e^7x using proofs. To show this, we will use the rules of differentiation. There are several methods we use to prove this, such as:
Using Chain Rule We will now demonstrate that the differentiation of e^7x results in 7e^7x
using the aforementioned method: Using Chain Rule To prove the differentiation of e^7x using the chain rule, We use the formula: Let u = 7x, then f(x) = e^u
By the chain rule: d/dx [e^u] = e^u · du/dx
Since du/dx = 7, substitute back: d/dx (e^7x) = e^7x · 7 = 7e^7x
Hence, proved.
Higher-Order Derivatives of e^7x
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^7x.
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^7x, we generally use fⁿ(x) for the nth derivative of a function f(x), which tells us the change in the rate of change.
Special Cases:
The exponential function e^7x has no points of discontinuity, so it is differentiable everywhere. When x = 0, the derivative of e^7x is 7e^0 = 7.
Common Mistakes and How to Avoid Them in Derivatives of e^7x
Students frequently make mistakes when differentiating e^7x. 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 correctly
Students may forget to apply the chain rule correctly, which can lead to incorrect results. Ensure that each step is written in order, and remember that the derivative of e^u is e^u times the derivative of u. This is particularly important when u is a function of x, like 7x.
Mistake 2
Confusing e^x and e^7x
Students might not remember that e^7x involves multiplying the derivative of the exponent, unlike e^x, which is its own derivative. Keep in mind that for e^7x, you must multiply by the derivative of 7x, which is 7.
Mistake 3
Incorrect use of the exponential function property
While differentiating exponential functions, students may misapply the properties of exponents.
For example, they might incorrectly differentiate e^7x as e^(7x) without using the chain rule.
Ensure the chain rule is applied correctly by multiplying by the derivative of the exponent.
Mistake 4
Not simplifying expressions
Students sometimes forget to simplify expressions, leading to incomplete or incorrect results. They often skip steps and directly arrive at the result. Ensure that each step is simplified in order, as it is important to avoid errors in the process.
Mistake 5
Not recognizing the domain of e^7x
The function e^7x is defined for all real numbers, but students sometimes forget its domain when differentiating. Remember that e^7x is differentiable everywhere, so the derivative is applicable for all real x.
Examples Using the Derivative of e^7x
Problem 1
Calculate the derivative of e^7x · sin(x)
Here, we have f(x) = e^7x · sin(x). Using the product rule, f'(x) = u′v + uv′ In the given equation, u = e^7x and v = sin(x).
Let’s differentiate each term, u′ = d/dx (e^7x) = 7e^7x v′ = d/dx (sin(x)) = cos(x)
Substituting into the given equation, f'(x) = (7e^7x) · sin(x) + (e^7x) · cos(x)
Let’s simplify terms to get the final answer, f'(x) = 7e^7x sin(x) + e^7x cos(x)
Thus, the derivative of the specified function is 7e^7x sin(x) + e^7x cos(x).
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 has a sales model represented by the function S(x) = e^7x, where S represents sales over time x. Measure the rate of change in sales when x = 1.
We have S(x) = e^7x (sales model)...(1)
Now, we will differentiate the equation (1) Take the derivative e^7x: dS/dx = 7e^7x
Given x = 1 (substitute this into the derivative) dS/dx = 7e^7(1) = 7e^7
Hence, at x = 1, the rate of change in sales is 7e^7.
Explanation
We find the rate of change of sales at x=1, which means that at this point, sales increase at a rate of 7e^7.
Problem 3
Derive the second derivative of the function y = e^7x.
The first step is to find the first derivative, dy/dx = 7e^7x...(1)
Now we will differentiate equation (1) to get the second derivative: d²y/dx² = d/dx [7e^7x] = 7 · 7e^7x = 49e^7x
Therefore, the second derivative of the function y = e^7x is 49e^7x.
Explanation
We use the step-by-step process, where we start with the first derivative, then differentiate again to find the second derivative. We simplify the terms to find the final answer.
Problem 4
Prove: d/dx (e^(14x)) = 14e^(14x).
Let’s start using the chain rule: Consider y = e^(14x).
To differentiate, we use the chain rule: dy/dx = e^(14x) · d/dx (14x)
Since the derivative of 14x is 14, dy/dx = e^(14x) · 14 = 14e^(14x) Hence proved.
Explanation
In this step-by-step process, we used the chain rule to differentiate the equation. Then, we replace 14x with its derivative. As a final step, we simplify to derive the equation.
Problem 5
Solve: d/dx (e^7x/x)
To differentiate the function, we use the quotient rule: d/dx (e^7x/x) = (d/dx (e^7x) · x - e^7x · d/dx(x))/x²
We will substitute d/dx (e^7x) = 7e^7x and d/dx (x) = 1 = (7e^7x · x - e^7x · 1)/x² = (7xe^7x - e^7x)/x² = e^7x (7x - 1)/x²
Therefore, d/dx (e^7x/x) = e^7x (7x - 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 e^7x
1.Find the derivative of e^7x.
2.Can we use the derivative of e^7x in real life?
3.Is it possible to take the derivative of e^7x at any point?
4.What rule is used to differentiate e^7x/x?
5.Are the derivatives of e^7x and e^x the same?
6.Can we find the derivative of the e^7x formula?
Important Glossaries for the Derivative of e^7x
- Derivative: The derivative of a function indicates how the given function changes in response to a slight change in x.
- Exponential Function: A mathematical function involving an exponent, frequently appearing in growth and decay problems.
- Chain Rule: A fundamental rule in calculus used to differentiate composite functions.
- Natural Exponential Function: The exponential function with base e, where e is approximately 2.718.
- Product Rule: A rule used to differentiate functions that are products of two or more functions.
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Jaskaran Singh Saluja
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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.
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