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108 LearnersLast updated on September 26, 2025
Derivative of 9e^x
We use the derivative of 9e^x, which is 9e^x, 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 9e^x in detail.
What is the Derivative of 9e^x?
We now understand the derivative of 9e^x. It is commonly represented as d/dx (9e^x) or (9e^x)', and its value is 9e^x. The function 9e^x has a clearly defined derivative, indicating it is differentiable for all x.
The key concepts are mentioned below: Exponential Function: (e^x is the base of natural logarithms).
Constant Multiple Rule: Rule for differentiating 9e^x (since it involves a constant multiple).
Exponential Rule: The derivative of e^x is e^x.
Derivative of 9e^x Formula
The derivative of 9e^x can be denoted as d/dx (9e^x) or (9e^x)'.
The formula we use to differentiate 9e^x is: d/dx (9e^x) = 9e^x (or) (9e^x)' = 9e^x
The formula applies to all x.
Proofs of the Derivative of 9e^x
We can derive the derivative of 9e^x using proofs. To show this, we will use the rules of differentiation. There are several methods we use to prove this, such as:
- Using the Constant Multiple Rule
- Using the Exponential Rule
We will now demonstrate that the differentiation of 9e^x results in 9e^x using the above-mentioned methods:
Using the Constant Multiple Rule
To prove the differentiation of 9e^x using the constant multiple rule, We use the formula:
d/dx (c·f(x)) = c·f'(x), where c is a constant.
Here, c = 9 and f(x) = e^x.
The derivative of e^x is itself, (e^x)'.
Therefore, the derivative is: d/dx (9e^x) = 9(d/dx e^x) = 9e^x
Hence, proved.
Using the Exponential Rule
To prove the differentiation of 9e^x using the exponential rule, We use the formula: d/dx (e^x) = e^x
Here, the function is 9e^x.
Using the constant multiple rule, d/dx (9e^x) = 9(d/dx e^x) = 9e^x
Thus, the derivative of 9e^x is 9e^x.
Higher-Order Derivatives of 9e^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 9e^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 9e^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:
For exponential functions like 9e^x, there are no special points where the derivative is undefined. The derivative 9e^x is always 9e^x for any value of x.
Common Mistakes and How to Avoid Them in Derivatives of 9e^x
Students frequently make mistakes when differentiating 9e^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 skip steps and directly arrive at the result, especially when solving using the exponential rule. Ensure that each step is written in order. It may seem redundant, but it is important to avoid errors in the process.
Mistake 2
Incorrect Application of the Exponential Rule
Students sometimes misapply the exponential rule, thinking that the derivative of e^x is zero or another incorrect value. Remember that the derivative of e^x is e^x itself. Always apply the correct rule to avoid errors.
Mistake 3
Not Simplifying the Result
While differentiating, students might not simplify the expression, leading to a cluttered or incorrect final result. Always simplify your answers to their simplest form.
Mistake 4
Forgetting to Differentiate the Entire Expression
Students may forget to differentiate the entire expression, especially when it involves multiple terms. Always ensure that every part of the expression is differentiated accordingly.
Mistake 5
Misplacing Constants
There is a common mistake where students at times forget to multiply the constants properly. For example, they might incorrectly write d/dx (9e^x) as e^x. Remember to multiply the entire expression by the constant.
Examples Using the Derivative of 9e^x
Problem 1
Calculate the derivative of (9e^x · e^x)
Here, we have f(x) = 9e^x · e^x.
Using the product rule, f'(x) = u′v + uv′ In the given equation, u = 9e^x and v = e^x.
Let’s differentiate each term, u′= d/dx (9e^x) = 9e^x v′= d/dx (e^x) = e^x
Substituting into the given equation, f'(x) = (9e^x)·(e^x) + (9e^x)·(e^x)
Let’s simplify terms to get the final answer, f'(x) = 18e^2x
Thus, the derivative of the specified function is 18e^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 certain population is modeled by the function P(x) = 9e^x, where x represents time in years. If x = 2 years, find the rate of growth of the population.
We have P(x) = 9e^x (population growth)...(1)
Now, we will differentiate the equation (1)
Take the derivative of 9e^x: dP/dx = 9e^x Given x = 2 (substitute this into the derivative) dP/dx = 9e^2
Hence, we get the rate of growth of the population at x = 2 years as 9e^2.
Explanation
We find the rate of growth of the population at x = 2 years by differentiating the population function and substituting x = 2 into the derivative.
Problem 3
Derive the second derivative of the function y = 9e^x.
The first step is to find the first derivative, dy/dx = 9e^x...(1)
Now we will differentiate equation (1) to get the second derivative: d²y/dx² = d/dx [9e^x] d²y/dx² = 9e^x
Therefore, the second derivative of the function y = 9e^x is 9e^x.
Explanation
We use the step-by-step process, where we start with the first derivative. We then take the derivative again to find the second derivative, which results in the same exponential function.
Problem 4
Prove: d/dx (81e^x) = 81e^x.
Let’s start using the constant multiple rule: Consider y = 81e^x
To differentiate, we use the constant multiple rule: dy/dx = 81(d/dx e^x)
Since the derivative of e^x is e^x, dy/dx = 81e^x Hence proved.
Explanation
In this step-by-step process, we used the constant multiple rule to differentiate the equation. We replaced e^x with its derivative. As a final step, we multiplied by the constant to derive the equation.
Problem 5
Solve: d/dx (9e^x/x)
To differentiate the function, we use the quotient rule: d/dx (9e^x/x) = (d/dx (9e^x)·x - 9e^x·d/dx(x))/x²
We will substitute d/dx (9e^x) = 9e^x and d/dx (x) = 1 = (9e^x·x - 9e^x·1)/x² = (9xe^x - 9e^x)/x² = 9e^x(x - 1)/x²
Therefore, d/dx (9e^x/x) = 9e^x(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 9e^x
1.Find the derivative of 9e^x.
2.Can we use the derivative of 9e^x in real life?
3.Is it possible to take the derivative of 9e^x at any point?
4.What rule is used to differentiate 9e^x/x?
5.Are the derivatives of 9e^x and e^x the same?
Important Glossaries for the Derivative of 9e^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 of the form e^x, where e is the base of natural logarithms.
- Constant Multiple Rule: A rule stating that the derivative of a constant times a function is the constant times the derivative of the function.
- Exponential Rule: The derivative of e^x is e^x.
- Quotient Rule: A formula for finding the derivative of a quotient of two functions.
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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.
