101 LearnersLast updated on July 24th, 2025
Derivative of e6x
We use the derivative of e^(6x), which is 6e^(6x), as a measuring tool for 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 talk about the derivative of e^(6x) in detail.
What is the Derivative of e6x?
We now understand the derivative of e^(6x). It is commonly represented as d/dx (e^(6x)) or (e^(6x))', and its value is 6e^(6x). The function e^(6x) has a clearly defined derivative, indicating it is differentiable within its domain. The key concepts are mentioned below: Exponential Function: (e^(6x) is an exponential function with a base of e and an exponent of 6x). Chain Rule: Rule for differentiating e^(6x) (since it involves a linear function of x in the exponent). Natural Exponential Function: e^x, where e is an irrational constant approximately equal to 2.71828.
Derivative of e6x Formula
The derivative of e^(6x) can be denoted as d/dx (e^(6x)) or (e^(6x))'. The formula we use to differentiate e^(6x) is: d/dx (e^(6x)) = 6e^(6x) The formula applies to all x and is based on the chain rule for differentiation.
Proofs of the Derivative of e6x
We can derive the derivative of e^(6x) using proofs. To show this, we will use the rules of differentiation. There are several methods we use to prove this, such as: Using Chain Rule We will now demonstrate that the differentiation of e^(6x) results in 6e^(6x) using the above-mentioned method: Using Chain Rule To prove the differentiation of e^(6x) using the chain rule, We use the formula: Let u = 6x, then e^(6x) = e^u By chain rule: d/dx [e^u] = (d/du [e^u]) * (du/dx) The derivative of e^u with respect to u is e^u. Now, du/dx = 6 Thus, d/dx [e^(6x)] = e^(6x) * 6 = 6e^(6x). Hence, proved.
Higher-Order Derivatives of e6x
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 make it easier to understand functions like e^(6x). 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^(6x), 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
Since e^(6x) is defined for all real numbers, there are no points where its derivative is undefined. However, evaluating the derivative at specific values of x can yield interesting results. For instance: When x = 0, the derivative of e^(6x) = 6e^(6*0) = 6e^0 = 6. When x = 1, the derivative of e^(6x) = 6e^(6*1) = 6e^6.
Common Mistakes and How to Avoid Them in Derivatives of e6x
Students frequently make mistakes when differentiating e^(6x). 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 properly, which can lead to incomplete or incorrect results. They often skip the step of differentiating the inner function (6x) and directly write the derivative as e^(6x). 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 to multiply by the inner function's derivative
They might not remember that the derivative of e^(6x) involves multiplying by the derivative of 6x, which is 6. Without this factor, the derivative would be incomplete. Always check that you've multiplied by the derivative of the inner function.
Mistake 3
Misunderstanding the exponential function's properties
While differentiating functions involving e^(6x), students may misapply the properties of exponential functions. For example, they might incorrectly assume that the derivative of e^(6x) is simply 6e^(x), forgetting the exponent's linear component. Always remember that the derivative of e^(ax) is ae^(ax), where a is a constant.
Mistake 4
Not simplifying expressions
There is a common mistake that students at times forget to simplify expressions after differentiating. For example, they might leave the expression as 6e^(6x), not simplifying further if needed in context. Ensure you simplify where possible, especially when solving equations or computing values.
Mistake 5
Confusing the base of the exponential
Students often confuse the base of the exponential function, especially when working with other exponential bases. Remember that the base e is specific to these calculations and has unique properties unlike other bases.
Examples Using the Derivative of e6x
Problem 1
Calculate the derivative of (e^(6x)·cos(x))
Here, we have f(x) = e^(6x)·cos(x). Using the product rule, f'(x) = u′v + uv′ In the given equation, u = e^(6x) and v = cos(x). Let’s differentiate each term, u′ = d/dx (e^(6x)) = 6e^(6x) v′ = d/dx (cos(x)) = -sin(x) Substituting into the given equation, f'(x) = (6e^(6x))·(cos(x)) + (e^(6x))·(-sin(x)) Let’s simplify terms to get the final answer, f'(x) = 6e^(6x)cos(x) - e^(6x)sin(x) Thus, the derivative of the specified function is 6e^(6x)cos(x) - e^(6x)sin(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
AXB International School is examining the growth of a bacterial culture. The population is represented by the function N = e^(6x), where N is the number of bacteria and x is time in hours. If x = 2 hours, calculate the rate of growth of the population.
We have N = e^(6x) (population growth function)...(1) Now, we will differentiate the equation (1) Take the derivative e^(6x): dN/dx = 6e^(6x) Given x = 2 (substitute this into the derivative) dN/dx = 6e^(6*2) = 6e^12 Hence, the rate of growth of the population at x = 2 hours is 6e^12.
Explanation
We find the rate of growth of the population at x = 2 hours, which shows how rapidly the bacterial population is increasing at that specific time.
Problem 3
Derive the second derivative of the function N = e^(6x).
The first step is to find the first derivative, dN/dx = 6e^(6x)...(1) Now we will differentiate equation (1) to get the second derivative: d²N/dx² = d/dx [6e^(6x)] Here we use the chain rule, d²N/dx² = 6 * (6e^(6x)) d²N/dx² = 36e^(6x) Therefore, the second derivative of the function N = e^(6x) is 36e^(6x).
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^(12x)) = 12e^(12x).
Let’s start using the chain rule: Consider y = e^(12x) To differentiate, we use the chain rule: dy/dx = d/dx [e^(12x)] The derivative of e^(u) with respect to u is e^(u). Now, let u = 12x, so du/dx = 12. Thus, dy/dx = 12 * e^(12x) Hence proved: d/dx (e^(12x)) = 12e^(12x).
Explanation
In this step-by-step process, we used the chain rule to differentiate the equation. Then, we replace the exponent with its derivative. As a final step, we simplify to derive the equation.
Problem 5
Solve: d/dx (e^(6x)/x)
To differentiate the function, we use the quotient rule: d/dx (e^(6x)/x) = (d/dx (e^(6x))·x - e^(6x)·d/dx(x))/x² We will substitute d/dx (e^(6x)) = 6e^(6x) and d/dx (x) = 1 = (6e^(6x)·x - e^(6x)·1) / x² = (6xe^(6x) - e^(6x)) / x² Therefore, d/dx (e^(6x)/x) = (6xe^(6x) - e^(6x)) / 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 e6x
1.Find the derivative of e^(6x).
2.Can we use the derivative of e^(6x) in real life?
3.Is it possible to take the derivative of e^(6x) at any real number?
4.What rule is used to differentiate e^(6x)/x?
5.Are the derivatives of e^(6x) and e^(x) the same?
6.Can we find the derivative of the e^(6x) formula?
Important Glossaries for the Derivative of e6x
Derivative: The derivative of a function indicates how the given function changes in response to a slight change in x. Exponential Function: A function where the variable is in the exponent, such as e^(6x). Chain Rule: A rule in calculus for differentiating the composition of two or more functions. Natural Exponential Function: The function e^x, where e is the base of natural logarithms. Product Rule: A rule used for finding the derivative of the product of two functions.
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.
