100 LearnersLast updated on July 21st, 2025
Derivative of 3e^3x
We explore the derivative of the exponential function 3e^3x, which is used to analyze how the function changes with respect to x. Derivatives are crucial for applications such as calculating rates of change in various fields. We will discuss the derivative of 3e^3x in detail.
What is the Derivative of 3e^3x?
The derivative of 3e^3x is represented as d/dx (3e^3x) or (3e^3x)', and it is 9e^3x. This derivative indicates that the function is differentiable for all real numbers x.
The key concepts include:
Exponential Function: (e^x is the natural exponential function).
Constant Multiplication Rule: The derivative of a constant multiplied by a function is the constant times the derivative of the function.
Chain Rule: Used for differentiating composite functions like e^(3x).
Derivative of 3e^3x Formula
The derivative of 3e^3x can be denoted as d/dx (3e^3x) or (3e^3x)'. The formula for differentiating 3e^3x is: d/dx (3e^3x) = 9e^3x
This formula is valid for all real numbers x.
Proofs of the Derivative of 3e^3x
We can derive the derivative of 3e^3x using different methods. Here, we will use the chain rule to prove this:
Using the Chain Rule
To differentiate 3e^3x using the chain rule, consider the function f(x) = 3e^3x. We can rewrite this as g(h(x)), where g(u) = 3e^u and h(x) = 3x.
The derivative of g(u) = 3e^u is g'(u) = 3e^u, and the derivative of h(x) = 3x is h'(x) = 3.
By the chain rule: d/dx [g(h(x))] = g'(h(x)) · h'(x)
Substituting the derivatives: d/dx (3e^3x) = 3e^(3x) · 3 = 9e^3x.
Thus, the derivative of 3e^3x is 9e^3x.
Higher-Order Derivatives of 3e^3x
Higher-order derivatives are obtained by differentiating a function multiple times. For the function 3e^3x, the derivatives follow a recognizable pattern due to the nature of the exponential function.
First Derivative: f'(x) = 9e^3x
Second Derivative: f''(x) = d/dx [9e^3x] = 27e^3x
Third Derivative: f'''(x) = d/dx [27e^3x] = 81e^3x
For the nth derivative of 3e^3x, we write f^(n)(x) = 3^n * e^3x, showing how the rate of change increases exponentially.
Special Cases:
The derivative of 3e^3x remains defined for all real numbers x because the exponential function is continuous everywhere. There are no discontinuities or undefined points for this function.
Common Mistakes and How to Avoid Them in Derivatives of 3e^3x
Students often make errors when differentiating 3e^3x. Understanding the correct methods can help avoid these issues. Below are some common mistakes and how to resolve them:
Mistake 1
Ignoring the Chain Rule
A frequent mistake is neglecting to apply the chain rule when differentiating composite functions like e^3x. This omission leads to incorrect results. Always identify the inner and outer functions and apply the chain rule correctly.
Mistake 2
Forgetting to Multiply by the Derivative of the Inner Function
Students sometimes forget to multiply by the derivative of the inner function when applying the chain rule. In the case of 3e^3x, they might overlook multiplying by 3. Ensure that the derivative of the inner function is considered.
Mistake 3
Incorrect Constant Multiplication
Some students fail to correctly multiply the constant 3 with the derivative of e^3x. This results in an incorrect answer. Always multiply the entire expression by the constant factor.
Mistake 4
Overlooking the Exponential Nature
Misunderstanding the exponential nature of e^3x can lead to errors. Remember that the derivative of e^u is e^u times the derivative of u.
Mistake 5
Not Simplifying Fully
Failing to simplify the expression can lead to incomplete answers. After finding the derivative, ensure that the expression is simplified as much as possible.
Examples Using the Derivative of 3e^3x
Problem 1
Calculate the derivative of (3e^3x · x^2)
Here, we have f(x) = 3e^3x · x^2.
Using the product rule, f'(x) = u′v + uv′ In the given equation, u = 3e^3x and v = x^2.
Let’s differentiate each term, u′= d/dx (3e^3x) = 9e^3x v′= d/dx (x^2) = 2x
Substituting into the given equation, f'(x) = (9e^3x)·(x^2) + (3e^3x)·(2x)
Let’s simplify terms to get the final answer, f'(x) = 9e^3x · x^2 + 6e^3x · x
Thus, the derivative of the specified function is 9e^3x · x^2 + 6e^3x · 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 tracks its revenue with the formula R(x) = 3e^3x, where x represents time in years. Calculate the rate of change of revenue when x = 2 years.
We have R(x) = 3e^3x (revenue function)...(1)
Now, we will differentiate the equation (1) Take the derivative: dR/dx = 9e^3x Given x = 2 (substitute this into the derivative) dR/dx = 9e^(3*2) = 9e^6
Hence, the rate of change of revenue at x = 2 years is 9e^6.
Explanation
We find the rate of change of revenue at x = 2 years as 9e^6. This represents how rapidly the revenue is increasing at that specific time.
Problem 3
Derive the second derivative of the function y = 3e^3x.
The first step is to find the first derivative, dy/dx = 9e^3x...(1)
Now we will differentiate equation (1) to get the second derivative:
d^2y/dx^2 = d/dx [9e^3x] d^2y/dx^2 = 27e^3x
Therefore, the second derivative of the function
y = 3e^3x is 27e^3x.
Explanation
We use the step-by-step process, where we start with the first derivative. By differentiating again, we find the second derivative, showing how the rate of change is increasing.
Problem 4
Prove: d/dx (9e^(3x)) = 27e^(3x).
Let’s start using the chain rule: Consider y = 9e^(3x)
To differentiate, we use the chain rule: dy/dx = 9 * d/dx [e^(3x)]
Since the derivative of e^(3x) is 3e^(3x), dy/dx = 9 * 3e^(3x)
dy/dx = 27e^(3x)
Hence proved.
Explanation
In this step-by-step process, we used the chain rule to differentiate the equation. Then, we replace e^(3x) with its derivative, simplifying to derive the equation.
Problem 5
Solve: d/dx (3e^3x/x)
To differentiate the function, we use the quotient rule: d/dx (3e^3x/x) = (d/dx (3e^3x) · x - 3e^3x · d/dx(x))/ x^2
We will substitute d/dx (3e^3x) = 9e^3x and d/dx (x) = 1 = (9e^3x · x - 3e^3x · 1) / x^2 = (9e^3x · x - 3e^3x) / x^2
Therefore, d/dx (3e^3x/x) = (9e^3x · x - 3e^3x) / x^2
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 3e^3x
1.Find the derivative of 3e^3x.
2.Can we use the derivative of 3e^3x in real life?
3.What is the derivative of 3e^3x at x = 0?
4.What rule is used to differentiate 3e^3x?
5.Is the exponential function 3e^3x always increasing?
Important Glossaries for the Derivative of 3e^3x
- 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 or a constant multiplied by it, indicating rapid growth or decay.
- Chain Rule: A fundamental rule in calculus used to differentiate composite functions.
- Constant Multiplication Rule: A rule stating that the derivative of a constant multiplied by a function is the constant times the derivative of the function.
- Product Rule: A rule used to differentiate products of 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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