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100 LearnersLast updated on September 15, 2025
Derivative of e^(2/x)
The derivative of e^(2/x) provides insight into how the function changes with respect to slight variations in x. Derivatives are crucial in various real-life applications, such as calculating rates of change and modeling exponential growth. In this article, we delve into the derivative of e^(2/x) in detail.
What is the Derivative of e^(2/x)?
To understand the derivative of e^(2/x), we represent it as d/dx (e^(2/x)) or (e^(2/x))'.
Using the chain rule, the derivative is found to be -2/x² * e^(2/x). The function e^(2/x) is differentiable across its domain, and its derivative is crucial for analyzing exponential growth and decay processes. Key concepts include: -
Exponential Function: e^u, where u is a function of x.
Chain Rule: A differentiation technique used for composite functions.
Power Rule: Used for differentiating functions of the form x^n.
Derivative of e^(2/x) Formula
The derivative of e^(2/x) can be expressed as d/dx (e^(2/x)) or (e^(2/x))'.
The formula to differentiate e^(2/x) is: d/dx (e^(2/x)) = -2/x² * e^(2/x), This formula is applicable for all x ≠ 0.
Proofs of the Derivative of e^(2/x)
We can derive the derivative of e^(2/x) using various methods. Here are some approaches:
- By Chain Rule
- By Product Rule
Let's demonstrate the differentiation of e^(2/x) using the chain rule:
Using Chain Rule
To prove the differentiation of e^(2/x) using the chain rule, we set: f(x) = 2/x
Then e^(2/x) becomes e^f(x).
The differentiation process involves: g(x) = e^u, where u = 2/x
Differentiating e^u gives: g'(x) = e^u * u' u' = d/dx (2/x) = -2/x²
Thus, the derivative of e^(2/x) is: d/dx (e^(2/x)) = e^(2/x) * (-2/x²) = -2/x² * e^(2/x)
Higher-Order Derivatives of e^(2/x)
Higher-order derivatives are obtained by differentiating the first derivative successively. For example, the second derivative involves differentiating the first derivative of e^(2/x) again. Higher-order derivatives are significant in understanding the behavior and curvature of functions like e^(2/x).
For the first derivative, we write f′(x), which gives the rate of change of the function. The second derivative, f′′(x), is derived from the first derivative and provides insights into the concavity of the function.
The nth Derivative, denoted as fⁿ(x), describes the change in the rate of change and can be useful in various applications.
Special Cases:
When x = 0, the function e^(2/x) is undefined.
Therefore, its derivative cannot be evaluated at this point. For positive or negative values of x where x ≠ 0, the derivative of e^(2/x) is -2/x² * e^(2/x).
Common Mistakes and How to Avoid Them in Derivatives of e^(2/x)
Students often make errors when differentiating e^(2/x). Understanding the correct process is key to avoiding these mistakes. Here are some common errors and solutions:
Mistake 1
Incorrect Application of the Chain Rule
Some students fail to correctly apply the chain rule, leading to incorrect results. It's crucial to identify the inner function and its derivative accurately. Always ensure that the derivative of the inner function is included in the final result.
Mistake 2
Neglecting to Simplify the Derivative
Students may forget to simplify the derivative, resulting in incomplete or incorrect expressions. Always simplify the derivative expression to its simplest form to avoid confusion.
Mistake 3
Forgetting to Apply the Power Rule
When finding the derivative of 2/x, students may skip the application of the power rule. Remember that 2/x can be rewritten as 2x⁻¹, and its derivative involves applying the power rule.
Mistake 4
Mixing Up Exponential and Polynomial Derivatives
Some students confuse the derivatives of exponential functions with polynomial functions. Ensure you differentiate using the appropriate rules for exponential functions like e^(2/x).
Mistake 5
Not Accounting for Undefined Points
e^(2/x) is undefined at x = 0. Students should consider the domain of the function and recognize points where it does not exist.
Examples Using the Derivative of e^(2/x)
Problem 1
Calculate the derivative of e^(2/x) * x²
Let f(x) = e^(2/x) * x².
Using the product rule, f'(x) = u'v + uv' Here, u = e^(2/x) and v = x².
Differentiate each term: u' = -2/x² * e^(2/x) v' = 2x
Substitute these into the equation: f'(x) = (-2/x² * e^(2/x)) * x² + (e^(2/x)) * 2x
Simplify the terms: f'(x) = -2e^(2/x) + 2xe^(2/x)
Thus, the derivative is -2e^(2/x) + 2xe^(2/x).
Explanation
The derivative is calculated by dividing the function into parts and applying the product rule. Each part is differentiated separately and then combined for the final result.
Problem 2
A bacteria culture grows at a rate modeled by y = e^(2/x), where y is the population size and x is the time in hours. Find the rate of growth at x = 2 hours.
Given y = e^(2/x), Differentiate y with respect to x: dy/dx = -2/x² * e^(2/x)
Substitute x = 2 into the derivative: dy/dx = -2/(2²) * e^(2/2) = -2/4 * e = -1/2 * e
The rate of growth at x = 2 hours is -1/2 * e.
Explanation
By substituting x = 2 into the derivative, we determine the rate of growth of the bacteria culture at that point in time.
Problem 3
Find the second derivative of y = e^(2/x).
First, find the first derivative: dy/dx = -2/x² * e^(2/x)
Now differentiate again: d²y/dx² = d/dx [-2/x² * e^(2/x)]
Using the product rule: = [-2/x² * d/dx (e^(2/x)) + e^(2/x) * d/dx (-2/x²)] = [-2/x² * (-2/x² * e^(2/x)) + e^(2/x) * (4/x³)]
Simplify: = [4/x⁴ * e^(2/x) + 4/x³ * e^(2/x)] = 4/x⁴ * e^(2/x) * (1 + x)
Thus, the second derivative is 4/x⁴ * e^(2/x) * (1 + x).
Explanation
The second derivative is found by differentiating the first derivative, applying the product rule again, and simplifying the resulting expression.
Problem 4
Prove: d/dx (e^(4/x)) = -4/x² * e^(4/x).
Let y = e^(4/x). Using the chain rule: u = 4/x Then, y = e^u.
The derivative is: dy/dx = e^u * du/dx du/dx = -4/x²
Therefore, dy/dx = e^(4/x) * (-4/x²) dy/dx = -4/x² * e^(4/x) Hence proved.
Explanation
The proof is accomplished by identifying the inner function and applying the chain rule to differentiate e^(4/x).
Problem 5
Solve: d/dx (x * e^(2/x))
Use the product rule: d/dx (x * e^(2/x)) = (d/dx (x) * e^(2/x) + x * d/dx (e^(2/x)))
Substitute d/dx (x) = 1 and d/dx (e^(2/x)) = -2/x² * e^(2/x): = (1 * e^(2/x) + x * (-2/x² * e^(2/x))) = e^(2/x) - 2/x * e^(2/x) = e^(2/x) * (1 - 2/x)
Therefore, the derivative is e^(2/x) * (1 - 2/x).
Explanation
The function is differentiated using the product rule, simplifying the expression to achieve the final result.
FAQs on the Derivative of e^(2/x)
1.What is the derivative of e^(2/x)?
2.Can the derivative of e^(2/x) be applied in real life?
3.Is the derivative of e^(2/x) defined at x = 0?
4.What rule is used to differentiate e^(2/x)?
5.Are the derivatives of e^(2/x) and ln(x) the same?
Important Glossaries for the Derivative of e^(2/x)
- Derivative: The rate at which a function changes with respect to a variable.
- Exponential Function: A function of the form e^u, where u is a variable expression.
- Chain Rule: A differentiation method for composite functions.
- Product Rule: A rule used to differentiate products of two functions.
- Undefined Point: A point where a function or its derivative does not exist or is not defined.
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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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