109 LearnersLast updated on August 5th, 2025
Derivative of e^(x+3)
We use the derivative of e^(x+3), which is e^(x+3), 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^(x+3) in detail.
What is the Derivative of e^(x+3)?
We now understand the derivative of e^(x+3). It is commonly represented as d/dx (e^(x+3)) or (e^(x+3))', and its value is e^(x+3). The function e^(x+3) has a clearly defined derivative, indicating it is differentiable for all real numbers.
The key concepts are mentioned below:
Exponential Function: e^(x+3) is an exponential function where the base is Euler's number.
Chain Rule: Rule used for differentiating composite functions like e^(x+3).
Derivative of e^x: The derivative of the natural exponential function e^x is itself, e^x.
Derivative of e^(x+3) Formula
The derivative of e^(x+3) can be denoted as d/dx (e^(x+3)) or (e^(x+3))'. The formula we use to differentiate e^(x+3) is: d/dx (e^(x+3)) = e^(x+3)
The formula applies to all x.
Proofs of the Derivative of e^(x+3)
We can derive the derivative of e^(x+3) using proofs. To show this, we will use the rules of differentiation. There are several methods we use to prove this, such as: U
- sing Chain Rule
- Using First Principles
Using Chain Rule
To prove the differentiation of e^(x+3) using the chain rule, Let u = x + 3, which makes e^(x+3) = e^u. Then, d/dx (e^(x+3)) = d/du (e^u) * du/dx
The derivative of e^u with respect to u is e^u and the derivative of u with respect to x is 1.
So, d/dx (e^(x+3)) = e^u * 1 = e^(x+3).
Using First Principles
The derivative of e^(x+3) can also be proved using the First Principle, which expresses the derivative as the limit of the difference quotient.
Consider f(x) = e^(x+3). Its derivative can be expressed as the following limit. f'(x) = limₕ→₀ [f(x + h) - f(x)] / h = limₕ→₀ [e^(x+h+3) - e^(x+3)] / h = limₕ→₀ e^(x+3) [e^h - 1] / h
Using the limit formula limₕ→₀ (e^h - 1)/h = 1, f'(x) = e^(x+3) * 1 = e^(x+3).
Hence, proved.
Higher-Order Derivatives of e^(x+3)
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+3).
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+3), we generally use f^(n)(x) for the nth derivative of a function f(x), which tells us the change in the rate of change.
Special Cases:
The derivative of e^(x+3) is always e^(x+3) regardless of x, as the exponential function is defined for all real numbers. However, at x = -3, e^(x+3) simplifies to e^0, which is 1.
Similarly, at any point, the derivative is simply the function value itself.
Common Mistakes and How to Avoid Them in Derivatives of e^(x+3)
Students frequently make mistakes when differentiating e^(x+3). 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 composite functions like e^(x+3). This can lead to incorrect results. Remember to differentiate the inner function (x+3) and multiply it with the derivative of the outer function e^u.
Mistake 2
Forgetting the Constant Multiple
They might not multiply the constant when differentiating functions like 5e^(x+3). Ensure that you multiply the constant by the derivative of the exponential function.
Mistake 3
Incorrectly Simplifying the Expression
While differentiating, students might incorrectly simplify e^(x+3) or forget to multiply the derivative of x+3 when using the chain rule. Ensure each step is correctly simplified and the chain rule is applied properly.
Mistake 4
Confusing Exponential Base
There is a common mistake where students confuse the base of the exponential function. Remember that e^(x+3) has a base of e, not a different number. Ensure you use the correct base when differentiating.
Mistake 5
Neglecting Exponential Properties
Some students might neglect the properties of exponentials, such as e^(a+b) = e^a * e^b, which could lead to incorrect differentiation. Always apply exponential properties correctly.
Examples Using the Derivative of e^(x+3)
Problem 1
Calculate the derivative of (e^(x+3) * ln(x))
Here, we have f(x) = e^(x+3) * ln(x).
Using the product rule, f'(x) = u′v + uv′ In the given equation, u = e^(x+3) and v = ln(x).
Let’s differentiate each term, u′ = d/dx (e^(x+3)) = e^(x+3) v′ = d/dx (ln(x)) = 1/x
Substituting into the given equation, f'(x) = (e^(x+3))(1/x) + (e^(x+3))(ln(x))
Let’s simplify terms to get the final answer, f'(x) = e^(x+3)/x + e^(x+3)ln(x)
Thus, the derivative of the specified function is e^(x+3)/x + e^(x+3)ln(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 models its revenue growth with the function R(x) = e^(x+3) where x is time in years. Find the rate of revenue growth at x = 2 years.
We have R(x) = e^(x+3) (revenue growth function)...(1)
Now, we will differentiate the equation (1)
Take the derivative e^(x+3): dR/dx = e^(x+3) Given x = 2, substitute this into the derivative, dR/dx = e^(2+3) = e^5
Hence, we get the rate of revenue growth at x = 2 years as e^5.
Explanation
We find the rate of revenue growth at x = 2 years by substituting x = 2 into the derivative of the given function. This gives us the rate of change of revenue at that specific time.
Problem 3
Derive the second derivative of the function y = e^(x+3).
The first step is to find the first derivative, dy/dx = e^(x+3)...(1)
Now, we will differentiate equation (1) to get the second derivative: d²y/dx² = d/dx [e^(x+3)]
Since the derivative of e^(x+3) is e^(x+3), d²y/dx² = e^(x+3)
Therefore, the second derivative of the function y = e^(x+3) is e^(x+3).
Explanation
Using the step-by-step process, we start with the first derivative. Since the derivative of an exponential function remains the same, the second derivative is also e^(x+3).
Problem 4
Prove: d/dx (e^(2x+6)) = 2e^(2x+6).
Let’s start using the chain rule: Consider y = e^(2x+6) We use the chain rule: dy/dx = d/du (e^u) * du/dx where u = 2x+6
The derivative of e^u with respect to u is e^u and the derivative of u with respect to x is 2. dy/dx = e^(2x+6) * 2
Thus, d/dx (e^(2x+6)) = 2e^(2x+6).
Hence proved.
Explanation
In this step-by-step process, we use the chain rule to differentiate the equation. We differentiate the inner function and multiply it with the derivative of the outer function to derive the equation.
Problem 5
Solve: d/dx (e^(x+3)/x)
To differentiate the function, we use the quotient rule: d/dx (e^(x+3)/x) = (d/dx (e^(x+3)).x - e^(x+3).d/dx(x))/x²
We will substitute d/dx (e^(x+3)) = e^(x+3) and d/dx (x) = 1 = (e^(x+3).x - e^(x+3))/x² = e^(x+3)(x - 1)/x²
Therefore, d/dx (e^(x+3)/x) = e^(x+3)(x - 1)/x².
Explanation
In this process, we differentiate the given function using the product rule and quotient rule. As a final step, we simplify the equation to obtain the final result.
FAQs on the Derivative of e^(x+3)
1.Find the derivative of e^(x+3).
2.Can we use the derivative of e^(x+3) in real life?
3.Is the derivative of e^(x+3) different from e^(x)?
4.What rule is used to differentiate e^(x+3)/x?
5.Can the derivative of e^(x+3) be used to find acceleration?
6.Is the derivative of e^(x+3) always positive?
Important Glossaries for the Derivative of e^(x+3)
- 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+3), where e is Euler's number, representing continuous growth or decay.
- Chain Rule: A rule in calculus for differentiating compositions of functions, such as e^(x+3).
- Quotient Rule: A rule for differentiating functions that are divided by each other, like e^(x+3)/x.
- Higher-Order Derivatives: Derivatives obtained by differentiating a function multiple times, such as the second or third derivative.
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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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