100 LearnersLast updated on July 17th, 2025
Derivative of e^6x
The derivative of e^6x provides insights into how the function changes as x changes. Derivatives are useful in various real-life applications, such as determining rates of change and growth. Here, we will delve into the derivative of e^6x in detail.
What is the Derivative of e^6x?
We now explore the derivative of e^6x, which is represented as d/dx (e^6x) or (e^6x)'. The derivative of e^6x is 6e^6x, illustrating that the function is differentiable across its domain.
Key concepts include: -
Exponential Function: e^x is the base function where e is the constant approximately equal to 2.71828.
Chain Rule: This rule is applied because of the inner function 6x.
Constant Multiplication: The presence of a constant multiplier in the exponent affects the derivative.
Derivative of e^6x Formula
The derivative of e^6x can be denoted as d/dx (e^6x) or (e^6x)'. To differentiate e^6x, we use the formula: d/dx (e^6x) = 6e^6x This formula applies to all x as the exponential function is defined for all real numbers.
Proofs of the Derivative of e^6x
We can derive the derivative of e^6x using different proofs. To demonstrate, we use differentiation rules such as the chain rule.
Here's how it's done: Using Chain Rule To prove the differentiation of e^6x
using the chain rule: Consider the function f(x) = e^6x. Apply the chain rule, which states that d/dx [f(g(x))] = f'(g(x)) · g'(x).
Here, f(x) = e^x and g(x) = 6x, so: d/dx (e^6x) = e^6x · d/dx (6x) = e^6x · 6 = 6e^6x.
Hence, the derivative of e^6x is 6e^6x, as proved using the chain rule.
Higher-Order Derivatives of e^6x
Higher-order derivatives involve differentiating a function multiple times. They provide insights into the behavior of functions beyond their initial rates of change. For e^6x: -
The first derivative, f'(x), indicates the rate of change of e^6x, which is 6e^6x.
The second derivative, f''(x), is the derivative of 6e^6x, resulting in 36e^6x.
Higher-order derivatives follow this pattern, with the nth derivative of e^6x being 6^n e^6x.
Special Cases
There are no undefined points for the derivative of e^6x, as the exponential function is continuous everywhere. However, at x = 0, the derivative is 6e^0, which equals 6.
Common Mistakes and How to Avoid Them in Derivatives of e^6x
Students often make errors when differentiating e^6x. Understanding the correct procedures helps avoid these mistakes. Below are common errors and solutions:
Mistake 1
Misapplying the Chain Rule
Sometimes, students forget to apply the chain rule when differentiating e^6x. They might mistakenly write the derivative as e^6x instead of 6e^6x. Remember, the chain rule requires multiplying by the derivative of the exponent, 6x, which is 6.
Mistake 2
Confusing Constants in the Exponent
A common mistake is ignoring the constant multiplier in the exponent. For example, mistakenly writing the derivative of e^6x as e^6x instead of 6e^6x. Always account for the constant when applying the chain rule.
Mistake 3
Incorrect Use of Exponential Rules
Students may incorrectly apply exponential rules, such as treating e^6x as a power function rather than an exponential function. This can lead to incorrect derivatives. Remember that the derivative of e^u, where u is a function of x, is e^u multiplied by the derivative of u.
Mistake 4
Not Simplifying Results
After differentiation, students might not fully simplify the result. For example, leaving the result as 6e^6x without realizing it can be further simplified (if applicable). Always check for simplifications after deriving.
Mistake 5
Overlooking the Exponential Nature
In some cases, students might treat e^6x as if it were a polynomial, leading to incorrect differentiation. Recognize that e^6x is an exponential function and must be treated accordingly.
Examples Using the Derivative of e^6x
Problem 1
Calculate the derivative of (e^6x · x^3).
Here, we have f(x) = e^6x · x^3. Using the product rule, f'(x) = u'v + uv' In the given equation, u = e^6x and v = x^3.
Let's differentiate each term: u' = d/dx (e^6x) = 6e^6x v' = d/dx (x^3) = 3x^2
Substituting into the product rule, f'(x) = (6e^6x)(x^3) + (e^6x)(3x^2) = 6x^3e^6x + 3x^2e^6x
Thus, the derivative of the specified function is 6x^3e^6x + 3x^2e^6x.
Explanation
We find the derivative of the given function by breaking it into two parts. First, find the derivative of each part and then combine them using the product rule for the final result.
Problem 2
A bank models the growth of an investment with the function V(x) = e^6x, where V represents the value of the investment and x is the time in years. Calculate the rate of growth when x = 2 years.
We have V(x) = e^6x, representing the investment growth. To find the rate of growth, differentiate V(x): dV/dx = 6e^6x.
Substitute x = 2 into the derivative: dV/dx = 6e^(6*2) = 6e^12.
Therefore, the rate of growth of the investment at x = 2 years is 6e^12.
Explanation
We determine the rate of growth by differentiating the function and substituting the given value of x. This provides the instantaneous rate of change of the investment value at x = 2 years.
Problem 3
Derive the second derivative of the function y = e^6x.
First, find the first derivative: dy/dx = 6e^6x.
Now, differentiate the first derivative to find the second derivative: d²y/dx² = d/dx (6e^6x) = 6 · 6e^6x = 36e^6x.
Therefore, the second derivative of the function y = e^6x is 36e^6x.
Explanation
We begin with the first derivative and then differentiate again to find the second derivative. The result shows the second rate of change for the function y = e^6x.
Problem 4
Prove: d/dx (e^12x) = 12e^12x.
Let’s use the chain rule: Consider y = e^12x.
Differentiate using the chain rule: dy/dx = e^12x · d/dx (12x) = e^12x · 12 = 12e^12x.
Hence, proved that d/dx (e^12x) = 12e^12x.
Explanation
In this step-by-step process, we used the chain rule to differentiate the equation, accounting for the constant multiplier in the exponent. The final result confirms the derivative of e^12x.
Problem 5
Solve: d/dx (e^6x/x).
To differentiate the function, use the quotient rule: d/dx (e^6x/x) = (d/dx (e^6x) · x - e^6x · d/dx (x))/x²
Substitute d/dx (e^6x) = 6e^6x and d/dx (x) = 1: = (6e^6x · x - e^6x · 1)/x² = (6xe^6x - e^6x)/x² = e^6x(6x - 1)/x².
Therefore, d/dx (e^6x/x) = e^6x(6x - 1)/x².
Explanation
In this process, we differentiate the function using the quotient rule and substitute the derivatives of each part. Finally, simplify the expression for the result.
FAQs on the Derivative of e^6x
1.Find the derivative of e^6x.
2.Can the derivative of e^6x be applied 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^6 the same?
6.How do you prove the derivative of e^6x?
Important Glossaries for the Derivative of e^6x
- Derivative: Indicates how a function changes as its input changes.
- Exponential Function: A function where a constant base is raised to a variable exponent, denoted as e^x.
- Chain Rule: A rule for finding the derivative of a composite function.
- Higher-Order Derivative: Derivatives taken multiple times, indicating successive rates of change.
- Product Rule: A rule for differentiating products 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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