103 LearnersLast updated on July 16th, 2025
Derivative of e^-3x
We use the derivative of e^-3x, which is -3e^-3x, to understand how the exponential function changes with a small change in x. Derivatives help us in various applications, including calculating rates of change in scientific and engineering contexts. We will now discuss the derivative of e^-3x in detail.
What is the Derivative of e^-3x?
We now understand the derivative of e^-3x. It is commonly represented as d/dx (e^-3x) or (e^-3x)', and its value is -3e^-3x. The function e^-3x has a well-defined derivative, indicating it is differentiable for all real numbers.
The key concepts are mentioned below:
Exponential Function: e^x, where e is the base of the natural logarithm.
Chain Rule: A rule used for differentiating compositions of functions.
Negative Exponent: Represents the reciprocal of the base raised to the positive exponent.
Derivative of e^-3x Formula
The derivative of e^-3x can be denoted as d/dx (e^-3x) or (e^-3x)'.
The formula we use to differentiate e^-3x is: d/dx (e^-3x) = -3e^-3x
This formula holds for all x.
Proofs of the Derivative of e^-3x
We can derive the derivative of e^-3x using different proofs. To demonstrate this, we will use differentiation rules, particularly focusing on the chain rule.
Here are some methods to prove this derivative: By Chain Rule To prove the differentiation of e^-3x
using the chain rule: Consider f(x) = e^u where u = -3x.
The derivative of e^u is e^u du/dx. Here, du/dx = -3. So, d/dx (e^-3x) = e^-3x * (-3) = -3e^-3x.
Hence, proved.
Higher-Order Derivatives of e^-3x
When a function is differentiated multiple times, the derivatives obtained are referred to as higher-order derivatives. Higher-order derivatives can be complex. To understand them better, think of velocity (first derivative) and acceleration (second derivative) in physics.
Higher-order derivatives help us analyze functions like e^-3x. 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, 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^-3x, 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:
For all x, the derivative of e^-3x is defined. When x is a large negative number, the exponential function e^-3x approaches zero, and its derivative also approaches zero.
Common Mistakes and How to Avoid Them in Derivatives of e^-3x
Students frequently make mistakes when differentiating e^-3x. These mistakes can be avoided by understanding the correct methods. Here are some 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. It is crucial to recognize the inner function and differentiate it properly. Ensure that each step of the chain rule is followed precisely.
Mistake 2
Ignoring the constant multiplier
Sometimes students ignore the constant multiplier that results from the chain rule. For e^-3x, the derivative is -3e^-3x, not just e^-3x. Always remember to multiply by the derivative of the inner function.
Mistake 3
Misinterpreting the base of the exponential function
Students might confuse different bases, like mistaking e for another number. Ensure that the base e, approximately 2.718, is used correctly in all calculations.
Mistake 4
Forgetting negative exponents
Students sometimes mismanage negative exponents, leading to incorrect simplifications. Remember that e^-3x is the reciprocal of e^3x, and handle exponents carefully.
Mistake 5
Incorrect notation
Errors in notation, such as forgetting parentheses or misplacing negative signs, can lead to mistakes. Always write expressions clearly and check for errors in notation to avoid confusion.
Examples Using the Derivative of e^-3x
Problem 1
Calculate the derivative of (e^-3x · x²)
Here, we have f(x) = e^-3x · x².
Using the product rule, f'(x) = u′v + uv′ In the given equation, u = e^-3x and v = x².
Let's differentiate each term, u′ = d/dx (e^-3x) = -3e^-3x v′ = d/dx (x²) = 2x
Substituting into the given equation, f'(x) = (-3e^-3x) · x² + e^-3x · (2x)
Let's simplify terms to get the final answer, f'(x) = -3x²e^-3x + 2xe^-3x
Thus, the derivative of the specified function is -3x²e^-3x + 2xe^-3x.
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 scientist is measuring the decay of a radioactive substance, represented by the function y = e^-3x, where y is the remaining quantity at time x. Calculate the rate of decay when x = 1.
We have y = e^-3x (decay function)...(1)
Now, we will differentiate the equation (1) Take the derivative of e^-3x: dy/dx = -3e^-3x
Given x = 1 (substitute this into the derivative)
dy/dx = -3e^-3(1) = -3e^-3 The rate of decay at x = 1 is -3e^-3.
Explanation
We find the rate of decay at x = 1 by taking the derivative of the decay function.
Substituting x = 1 into the derivative gives us the rate at that point.
Problem 3
Derive the second derivative of the function y = e^-3x.
The first step is to find the first derivative, dy/dx = -3e^-3x...(1)
Now we will differentiate equation (1) to get the second derivative: d²y/dx² = d/dx [-3e^-3x]
d²y/dx² = (-3) * (-3e^-3x)
d²y/dx² = 9e^-3x
Therefore, the second derivative of the function y = e^-3x is 9e^-3x.
Explanation
We use a step-by-step process, where we start with the first derivative. We then differentiate it again and simplify to find the second derivative.
Problem 4
Prove: d/dx (e^-3x · e^x) = -2e^-2x.
Let's start using the product rule: Consider y = e^-3x · e^x
To differentiate, we use the product rule: dy/dx = (d/dx [e^-3x]) · e^x + e^-3x · (d/dx [e^x])
The derivatives are: d/dx [e^-3x] = -3e^-3x d/dx [e^x] = e^x
Substitute the derivatives: dy/dx = (-3e^-3x) · e^x + e^-3x · e^x = -3e^-2x + e^-2x = -2e^-2x
Hence proved.
Explanation
In this step-by-step process, we used the product rule to differentiate the equation. We then simplify by combining like terms to derive the equation.
Problem 5
Solve: d/dx (e^-3x/x)
To differentiate the function, we use the quotient rule: d/dx (e^-3x/x) = (d/dx [e^-3x] · x - e^-3x · d/dx [x]) / x²
We will substitute d/dx [e^-3x] = -3e^-3x and d/dx [x] = 1 = (-3e^-3x · x - e^-3x · 1) / x² = (-3xe^-3x - e^-3x) / x² = -e^-3x(3x + 1) / x²
Therefore, d/dx (e^-3x/x) = -e^-3x(3x + 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^-3x
1.Find the derivative of e^-3x.
2.Can we use the derivative of e^-3x in real life?
3.What rule is used to differentiate e^-3x/x?
4.Are the derivatives of e^-3x and e^3x the same?
5.What is the second derivative of e^-3x?
Important Glossaries for the Derivative of e^-3x
- Derivative: The derivative of a function indicates how the given function changes in response to a small change in x.
- Exponential Function: A mathematical function in the form of e^x, where e is the base of natural logarithms.
- Chain Rule: A rule for differentiating the composition of functions.
- Product Rule: A rule used to differentiate products of two functions.
- Quotient Rule: A rule used to differentiate the quotient of two 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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