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104 LearnersLast updated on October 3, 2025
Derivative of e^2x^2
We use the derivative of e^2x^2 to understand how this exponential function changes in response to a slight change in x. Derivatives are instrumental in calculating changes in various real-life situations, such as growth rates. We will now discuss the derivative of e^2x^2 in detail.
What is the Derivative of e^2x^2?
We now understand the derivative of\( e^2x^2\). It is commonly represented as\( d/dx (e^2x^2)\) or \((e^2x^2)'\), and its value is \(4xe^2x^2\). The function\( e^2x^2\) has a clearly defined derivative, indicating it is differentiable at all points. The key concepts are mentioned below:
Exponential Function: (\(e^2x^2\) = e raised to the power of\( 2x^2\)).
Chain Rule: Rule for differentiating composite functions like \(e^2x^2\).
Constant Multiple Rule: Used when differentiating a function multiplied by a constant.
Derivative of e^2x^2 Formula
The derivative of\( e^2x^2\) can be denoted as d/dx \((e^2x^2\)) or \((e^2x^2\)').
The formula for differentiating \(e^2x^2 \)is: d/dx \((e^2x^2)\) = \(4xe^2x^2 \)
The formula applies to all x in the domain of real numbers.
Proofs of the Derivative of e^2x^2
We can derive the derivative of e^2x^2 using proofs. To show this, we will use differentiation rules such as the chain rule. There are several methods we use to prove this, such as:
- By First Principle
- Using Chain Rule
We will now demonstrate that the differentiation of \(4xe^2x^2 \)results in\( 4xe^2x^2 \)using these methods:
Using Chain Rule
To prove the differentiation of e^2x^2 using the chain rule, Consider f(x) =\( 4xe^2x^2 \), and let g(u) = \(e^u\) where u = \(2x^2\).
According to the chain rule, d/dx [g(f(x))] = g'(f(x)) * f'(x).
First, find the derivative of f(x): f'(x) = d/dx (\(2x^2\)) = 4x.
Next, find the derivative of g(u): g'(u) = e^u. Substituting back, we have:
\(d/dx\) \([e^2x^2]\) = \(e^2x^2 * 4x. d/dx [e^2x^2] = 4xe^2x^2\).
Hence, proved.
Higher-Order Derivatives of e^2x^2
When a function is differentiated several times, the derivatives obtained are referred to as higher-order derivatives. Higher-order derivatives can become more complex.
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 make it easier to analyze functions like e^2x^2.
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^2x^2, we generally use f^(n)(x) to denote 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:
When x is any real number, the derivative is always defined because\( e^2x^2\) is defined for all real numbers. When x is 0, the derivative of \(e^2x^2 = 4x e^2x^2\), which evaluates to 0.
Common Mistakes and How to Avoid Them in Derivatives of e^2x^2
Students frequently make mistakes when differentiating\( e^2x^2\). These mistakes can be resolved by understanding the proper solutions. Here are a few common mistakes and ways to solve them:
Mistake 1
Not simplifying the equation
Students may forget to simplify the equation, which can lead to incomplete or incorrect results. They often skip steps and directly arrive at the result, especially when using the chain rule. Ensure that each step is written in order and fully simplified.
Mistake 2
Forgetting to apply the Chain Rule
Students might not remember to use the chain rule when differentiating functions that involve compositions like \(e^2x^2\). Always identify the inner and outer functions and apply the chain rule correctly.
Mistake 3
Incorrect use of Constant Multiple Rule
While differentiating functions, students may misapply the constant multiple rule. For example: Incorrect differentiation: \(d/dx (e^2x^2) = 2e^2x^2\). Remember that the derivative of \(e^u\) involves multiplying by the derivative of the exponent.
Mistake 4
Not writing Constants and Coefficients
There is a common mistake where students forget to multiply the constants placed before expressions. For example, they incorrectly write \(d/dx (5e^2x^2) = 4xe^2x^2\). Students should check the constants in the terms and ensure they are multiplied properly. For example, the correct equation is \(d/dx (5e^2x^2) = 20xe^2x^2.\)
Mistake 5
Confusing Exponential and Polynomial Differentiation
Students often confuse the differentiation rules for exponential functions with those for polynomials. Remember that the derivative of eu is eu times the derivative of u, whereas the power rule applies to polynomials.
Examples Using the Derivative of e^2x^2
Problem 1
Calculate the derivative of (e^2x^2 * ln x)
Here, we have f(x) =\( e^2x^2 \)* ln x. Using the product rule, f'(x) = u′v + uv′ In the given equation, u = \(e^2x^2\) and v = ln x. Let’s differentiate each term, \(u′ = d/dx (e^2x^2) = 4xe^2x^2 v′ = d/dx (ln x) = 1/x \)
Substituting into the given equation,\( f'(x) = (4xe^2x^2).(ln x) + (e^2x^2).(1/x)\) Let’s simplify terms to get the final answer,\( f'(x) = 4xe^2x^2 ln x + e^2x^2 / x\) Thus, the derivative of the specified function is \(4xe^2x^2 ln x + e^2x^2 / x.\)
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 the growth of a colony of bacteria over time, represented by the function y = e^2x^2, where y represents the number of bacteria at time x. If x = 1 hour, measure the rate of growth of the bacteria.
We have \(y = e^2x^2\) (growth model)...(1)
Now, we will differentiate the equation (1)
Take the derivative\( e^2x^2: dy/dx = 4xe^2x^2\) Given x = 1 (substitute this into the derivative)\( dy/dx = 4(1)e^(2(1)^2) = 4e^2 \)
Therefore, the rate of growth of the bacteria at x = 1 hour is \(4e^2\).
Explanation
We find the growth rate of the bacteria at x = 1 hour as \(4e^2\), which means that at one hour, the number of bacteria increases at a rate proportional to 4e^2.
Problem 3
Derive the second derivative of the function y = e^2x^2.
The first step is to find the first derivative, \(dy/dx = 4xe^2x^2.\)..(1)
Now we will differentiate equation (1) to get the second derivative:\( d²y/dx² = d/dx [4xe^2x^2] \)
Here we use the product rule,\( d²y/dx² = 4 * d/dx [xe^2x^2] = 4 * (e^2x^2 + 2x * 2xe^2x^2) = 4 * e^2x^2 (1 + 4x^2) \)
Therefore, the second derivative of the function \(y = e^2x^2 is 4e^2x^2 (1 + 4x^2).\)
Explanation
We use the step-by-step process, starting with the first derivative. By applying the product rule, we differentiate the function further, substituting and simplifying to find the final answer.
Problem 4
Prove: d/dx (e^(2x^2 + 3)) = 4xe^(2x^2 + 3).
Let’s start using the chain rule: Consider y = e(2x^2 + 3)
To differentiate, we use the chain rule: \(dy/dx = e^(2x^2 + 3) * d/dx [2x^2 + 3]\)
Since the derivative of\( 2x^2 + 3 is 4x, dy/dx = e^(2x^2 + 3) * 4x\)
Substituting y = \(e^(2x^2 + 3), d/dx (e^(2x^2 + 3)) = 4xe^(2x^2 + 3)
\)
hence proved!
Explanation
In this step-by-step process, we used the chain rule to differentiate the equation. We then replaced the inner function with its derivative and simplified to derive the equation.
Problem 5
Solve: d/dx (e^2x^2/x)
To differentiate the function, we use the quotient rule:\( d/dx (e^2x^2/x) = (d/dx (e^2x^2) * x - e^2x^2 * d/dx(x)) / x^2\)
We will substitute \(d/dx \)\((e^2x^2) = 4xe^2x^2 \) and d/dx (x) = 1 = \((4xe^2x^2 * x - e^2x^2 * 1) / x^2 = (4x^2e^2x^2 - e^2x^2) / x^2 = e^2x^2 (4x^2 - 1) / x^2\)
Therefore,\( d/dx (e^2x^2/x) = e^2x^2 (4x^2 - 1) / x^2\)
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 e^2x^2
1.Find the derivative of e^2x^2.
2.Can we use the derivative of e^2x^2 in real life?
3.Is it possible to take the derivative of e^2x^2 at any real number?
4.What rule is used to differentiate e^2x^2/x?
5.Are the derivatives of e^2x^2 and e^(2x^2 + 3) the same?
Important Glossaries for the Derivative of e^2x^2
- Derivative: The derivative of a function indicates how the given function changes in response to a slight change in x.
- Exponential Function: A mathematical function of the form e^u, where e is the base of the natural logarithm.
- Chain Rule: A rule in calculus for differentiating the composition of two or more functions.
- Constant Multiple Rule: A rule that allows taking the derivative of a constant multiplied by a function.
- Quotient Rule: A rule for finding the derivative of a 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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