101 LearnersLast updated on July 15th, 2025
Derivative of e^-4x
We use the derivative of e^-4x, which is -4e^-4x, as a tool for understanding 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 discuss the derivative of e^-4x in detail.
What is the Derivative of e^-4x?
We now understand the derivative of e^-4x. It is commonly represented as d/dx (e^-4x) or (e^-4x)', and its value is -4e^-4x. The function e^-4x has a clearly defined derivative, indicating it is differentiable for all real numbers. The key concepts are mentioned below: Exponential Function: (e^-4x is an exponential function). Chain Rule: Rule for differentiating composite functions (since e^-4x is a composite function of e^x).
Derivative of e^-4x Formula
The derivative of e^-4x can be denoted as d/dx (e^-4x) or (e^-4x)'. The formula we use to differentiate e^-4x is: d/dx (e^-4x) = -4e^-4x The formula applies to all real numbers x.
Proofs of the Derivative of e^-4x
We can derive the derivative of e^-4x using proofs. To show this, we will use the chain rule along with the rules of differentiation. There are several methods we use to prove this, such as: Using Chain Rule Using Chain Rule To prove the differentiation of e^-4x using the chain rule, We use the formula: Let u = -4x, then y = e^u. By chain rule: dy/dx = dy/du * du/dx dy/du = d/du (e^u) = e^u du/dx = d/dx (-4x) = -4 Substituting these into the chain rule, dy/dx = e^u * (-4) Since u = -4x, dy/dx = e^-4x * (-4) dy/dx = -4e^-4x Hence, proved.
Higher-Order Derivatives of e^-4x
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 help us understand functions like e^-4x better. 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^-4x, 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:
The function e^-4x is always differentiable for all real numbers x. When x = 0, the derivative of e^-4x is -4e^0, which equals -4.
Common Mistakes and How to Avoid Them in Derivatives of e^-4x
Students frequently make mistakes when differentiating e^-4x. 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. They often skip steps and directly arrive at the result, especially when dealing with composite functions. Ensure that each step is written in order. Students might think it is awkward, but it is important to avoid errors in the process.
Mistake 2
Forgetting the negative sign
They might not remember to apply the negative sign from the derivative of -4x. Keep in mind that you should include the negative sign when differentiating, as it is crucial to obtaining the correct derivative.
Mistake 3
Incorrect use of the Exponential Function Rule
While differentiating exponential functions, students might misapply the exponential function rule. For example: Incorrect differentiation: d/dx (e^-4x) = e^-4x. To avoid this mistake, always remember that the derivative of e^u is e^u * u'. Always check for errors in the calculation and ensure it is properly simplified.
Mistake 4
Not writing Constants and Coefficients
There is a common mistake where students at times forget to multiply the constants placed before e^-4x. For example, they incorrectly write d/dx (5e^-4x) = e^-4x. Students should check the constants in the terms and ensure they are multiplied properly. For example, the correct equation is d/dx (5e^-4x) = 5(-4e^-4x) = -20e^-4x.
Mistake 5
Confusing the Derivative with the Original Function
Students often confuse the derivative with the original function. This happens when they forget that the derivative represents the rate of change. For example: Incorrect: d/dx (e^-4x) = e^-4x. To fix this error, always differentiate the exponent and include it in the final derivative. For example, d/dx (e^-4x) = -4e^-4x.
Examples Using the Derivative of e^-4x
Problem 1
Calculate the derivative of e^-4x * e^3x
Here, we have f(x) = e^-4x * e^3x. Using the product rule, f'(x) = u′v + uv′ In the given equation, u = e^-4x and v = e^3x. Let’s differentiate each term, u′ = d/dx (e^-4x) = -4e^-4x v′ = d/dx (e^3x) = 3e^3x Substituting into the given equation, f'(x) = (-4e^-4x) * (e^3x) + (e^-4x) * (3e^3x) Let’s simplify terms to get the final answer, f'(x) = -4e^-x + 3e^-x f'(x) = (-4 + 3)e^-x f'(x) = -e^-x Thus, the derivative of the specified function is -e^-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 radioactive substance decays over time, and its quantity is represented by the function Q = e^-4x, where Q is the quantity remaining and x is the time in hours. Calculate the rate of decay when x = 1 hour.
We have Q = e^-4x (rate of decay)...(1) Now, we will differentiate the equation (1) Take the derivative of e^-4x: dQ/dx = -4e^-4x Given x = 1 (substitute this into the derivative) dQ/dx = -4e^-4(1) dQ/dx = -4e^-4 dQ/dx ≈ -0.0733 (since e^-4 ≈ 0.0183) Hence, we get the rate of decay at x = 1 hour as approximately -0.0733.
Explanation
We find the rate of decay at x = 1 hour to be approximately -0.0733, which means that at this point, the quantity is decreasing at this rate.
Problem 3
Derive the second derivative of the function Q = e^-4x.
The first step is to find the first derivative, dQ/dx = -4e^-4x...(1) Now we will differentiate equation (1) to get the second derivative: d²Q/dx² = d/dx [-4e^-4x] d²Q/dx² = -4 * d/dx [e^-4x] d²Q/dx² = -4 * (-4e^-4x) d²Q/dx² = 16e^-4x Therefore, the second derivative of the function Q = e^-4x is 16e^-4x.
Explanation
We use the step-by-step process, where we start with the first derivative. Using the chain rule, we differentiate again. We then simplify the terms to find the final answer.
Problem 4
Prove: d/dx (e^-8x) = -8e^-8x.
Let’s start using the chain rule: Consider y = e^-8x To differentiate, we use the chain rule: dy/dx = e^-8x * d/dx (-8x) Since the derivative of -8x is -8, dy/dx = e^-8x * (-8) Substituting y = e^-8x, d/dx (e^-8x) = -8e^-8x Hence proved.
Explanation
In this step-by-step process, we used the chain rule to differentiate the equation. Then, we replace -8x with its derivative. As a final step, we substitute y = e^-8x to derive the equation.
Problem 5
Solve: d/dx (e^-4x/x)
To differentiate the function, we use the quotient rule: d/dx (e^-4x/x) = (d/dx (e^-4x) * x - e^-4x * d/dx(x))/ x² We will substitute d/dx (e^-4x) = -4e^-4x and d/dx (x) = 1 = (-4e^-4x * x - e^-4x * 1) / x² = (-4xe^-4x - e^-4x) / x² = -e^-4x (4x + 1) / x² Therefore, d/dx (e^-4x/x) = -e^-4x (4x + 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^-4x
1.Find the derivative of e^-4x.
2.Can we use the derivative of e^-4x in real life?
3.Is it possible to take the derivative of e^-4x at any real number x?
4.What rule is used to differentiate e^-4x/x?
5.Are the derivatives of e^-4x and its reciprocal the same?
6.Can we find the derivative of the e^-4x formula?
Important Glossaries for the Derivative of e^-4x
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^kx, where k is a constant, such as e^-4x. Chain Rule: A rule for differentiating composite functions by differentiating the outer function and multiplying by the derivative of the inner function. Quotient Rule: A method for differentiating a function that is the quotient of two other functions. Exponential Decay: A decrease in quantity that follows an exponential function, such as e^-4x.
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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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