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102 LearnersLast updated on September 15, 2025
Derivative of e^-7x
We use the derivative of e^-7x, which is -7e^-7x, to understand how the exponential function changes in response to a slight change in x. Derivatives are crucial in calculating rates of change in real-life situations. We will now discuss the derivative of e^-7x in detail.
What is the Derivative of e^-7x?
We now understand the derivative of e^-7x. It is commonly represented as d/dx (e^-7x) or (e^-7x)', and its value is -7e^-7x. The function e^-7x has a clearly defined derivative, indicating it is differentiable within its domain. The key concepts are mentioned below:
Exponential Function: e^-7x is an exponential function with a base of e.
Chain Rule: Rule for differentiating composite functions.
Constant Multiple Rule: Enables differentiation of functions multiplied by a constant.
Derivative of e^-7x Formula
The derivative of e^-7x can be denoted as d/dx (e^-7x) or (e^-7x)'.
The formula we use to differentiate e^-7x is: d/dx (e^-7x) = -7e^-7x
The formula applies for all x, as the exponential function is continuous everywhere.
Proofs of the Derivative of e^-7x
We can derive the derivative of e^-7x using proofs. To show this, we will use the rules of differentiation, particularly the chain rule. Here is how we can prove this:
Using the Chain Rule
To prove the differentiation of e^-7x using the chain rule, We use the formula:
Let f(x) = e^u where u = -7x
Using the chain rule: d/dx (e^u) = e^u · du/dx
Therefore, d/dx (e^-7x) = e^-7x · d/dx(-7x) = e^-7x · (-7) = -7e^-7x
Hence, proved.
Higher-Order Derivatives of e^-7x
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, consider how acceleration (second derivative) is the rate of change of velocity (first derivative). Higher-order derivatives can reveal the behavior of functions like e^-7x.
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^-7x, 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 (continuing for higher-order derivatives).
Special Cases:
Since e^-7x is an exponential function, it is continuous everywhere, and its derivative is defined for all x. The derivative is always negative, indicating that the function is decreasing for all real numbers.
Common Mistakes and How to Avoid Them in Derivatives of e^-7x
Students frequently make mistakes when differentiating e^-7x. These mistakes can be resolved by understanding the correct methods. Here are a few common mistakes and ways to solve them:
Mistake 1
Not applying the chain rule correctly
A common mistake is neglecting the chain rule when differentiating composite functions. In the function e^-7x, ensure you differentiate the inner function -7x correctly and apply the chain rule to get -7e^-7x.
Mistake 2
Ignoring the negative sign
Students might forget the negative sign resulting from the derivative of -7x. Ensure that the negative sign is included in the derivative to avoid incorrect results.
Mistake 3
Forgetting the constant multiple rule
When differentiating, students might forget that constants are multiplied directly. For example, in differentiating e^-7x, -7 is a constant factor that should be correctly applied as -7e^-7x.
Mistake 4
Mistaking e^x for e^-7x
Confusion between different exponential forms is common. Remember that e^x and e^-7x have different derivatives, e^x and -7e^-7x, respectively.
Mistake 5
Misapplying differentiation rules
Students might apply incorrect differentiation rules. Ensure you understand which rule applies to e^-7x, primarily the chain rule for composite functions.
Examples Using the Derivative of e^-7x
Problem 1
Calculate the derivative of (e^-7x · sin(x))
Here, we have f(x) = e^-7x · sin(x).
Using the product rule, f'(x) = u′v + uv′ In the given equation, u = e^-7x and v = sin(x).
Let’s differentiate each term, u′ = d/dx (e^-7x) = -7e^-7x v′ = d/dx (sin(x)) = cos(x) Substituting into the given equation, f'(x) = (-7e^-7x) · sin(x) + e^-7x · cos(x)
Let’s simplify terms to get the final answer, f'(x) = -7e^-7x · sin(x) + e^-7x · cos(x)
Thus, the derivative of the specified function is -7e^-7x · sin(x) + e^-7x · cos(x).
Explanation
We find the derivative of the given function by dividing it into two parts. The first step is finding their derivatives 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, modeled by the function N(t) = e^-7t, where N(t) is the amount of substance remaining at time t. Find the rate of decay when t = 1 hour.
We have N(t) = e^-7t (decay of the substance)...(1)
Now, we will differentiate the equation (1) to find the rate of decay: dN/dt = -7e^-7t
Given t = 1 (substitute this into the derivative) dN/dt = -7e^-7(1) = -7e^-7
Hence, the rate of decay at t = 1 hour is -7e^-7.
Explanation
We find the rate of decay at t = 1 hour as -7e^-7, which represents the negative rate at which the substance decays over time.
Problem 3
Derive the second derivative of the function y = e^-7x.
The first step is to find the first derivative, dy/dx = -7e^-7x...(1)
Now we will differentiate equation (1) to get the second derivative:
d²y/dx² = d/dx [-7e^-7x] = -7 · d/dx [e^-7x] = -7(-7e^-7x) = 49e^-7x
Therefore, the second derivative of the function y = e^-7x is 49e^-7x.
Explanation
We use the step-by-step process, starting with the first derivative. By differentiating again, we find the second derivative using the constant multiple and chain rules.
Problem 4
Prove: d/dx (e^-7x²) = -14xe^-7x².
Let’s start using the chain rule: Consider y = e^-7x²
To differentiate, we use the chain rule: dy/dx = e^-7x² · d/dx (-7x²) = e^-7x² · (-14x) = -14xe^-7x²
Hence proved.
Explanation
In this step-by-step process, we used the chain rule to differentiate the equation. The derivative of the inner function -7x² is -14x, which we multiply with e^-7x² to get the result.
Problem 5
Solve: d/dx (e^-7x/x)
To differentiate the function, we use the quotient rule: d/dx (e^-7x/x) = (d/dx (e^-7x) · x - e^-7x · d/dx(x))/x²
We will substitute d/dx (e^-7x) = -7e^-7x and d/dx(x) = 1 = (-7e^-7x · x - e^-7x · 1)/x² = (-7xe^-7x - e^-7x)/x² = -7xe^-7x - e^-7x/x²
Therefore, d/dx (e^-7x/x) = (-7xe^-7x - e^-7x)/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^-7x
1.Find the derivative of e^-7x.
2.Can we use the derivative of e^-7x in real life?
3.Is the derivative of e^-7x always negative?
4.What rule is used to differentiate e^-7x/x?
5.Are the derivatives of e^-7x and e^x the same?
Important Glossaries for the Derivative of e^-7x
- Derivative: The derivative of a function indicates the rate at which the function changes in response to a change in x.
- Exponential Function: Functions that have a constant base raised to a variable power, such as e^x.
- Chain Rule: A rule used to differentiate composite functions.
- Constant Multiple Rule: Indicates how to differentiate functions that are multiplied by a constant.
- Quotient Rule: A rule used to differentiate functions that are divided by each other.
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.
