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109 LearnersLast updated on September 1, 2025
Derivative of n^x
We use the derivative of n^x, which is n^x ln(n), as a measuring tool for how the exponential function changes in response to a slight change in x. Derivatives help us calculate growth rates or decay in real-life situations. We will now talk about the derivative of n^x in detail.
What is the Derivative of n^x?
We now understand the derivative of nx.
It is commonly represented as d/dx (nx) or (nx)', and its value is nx ln(n).
The function nx has a clearly defined derivative, indicating it is differentiable within its domain.
The key concepts are mentioned below:
Exponential Function: (nx where n is a constant).
Chain Rule: Rule for differentiating nx.
Natural Logarithm: ln(n) is the natural logarithm of the base n.
Derivative of n^x Formula
The derivative of nx can be denoted as d/dx (nx) or (nx)'.
The formula we use to differentiate nx is: d/dx (nx) = nx ln(n) (or) (nx)' = nx ln(n)
The formula applies to all x where n > 0 and n ≠ 1.
Proofs of the Derivative of n^x
We can derive the derivative of nx using proofs.
To show this, we will use the logarithmic differentiation along with the rules of differentiation.
There are several methods we use to prove this, such as:
Using Logarithmic Differentiation
Using the Chain Rule
We will now demonstrate that the differentiation of nx results in nx ln(n) using the above-mentioned methods:
Using Logarithmic Differentiation
To find the derivative of nx using logarithmic differentiation, we consider f(x) = nx. Taking the natural log of both sides, we have ln(f(x)) = ln(nx) = x ln(n).
Differentiating both sides with respect to x, we obtain d/dx [ln(f(x))] = d/dx [x ln(n)] (1/f(x)) f'(x) = ln(n) f'(x) = f(x) ln(n) Substituting f(x) = nx, we get f'(x) = nx ln(n)
Hence, proved.
Using the Chain Rule
To prove the differentiation of nx using the chain rule, Consider f(x) = e(x ln(n))
By the chain rule, the derivative is: d/dx (e(x ln(n))) = e(x ln(n)) ·
d/dx (x ln(n)) = e(x ln(n)) ·
ln(n) Since e(x ln(n)) = nx,
we have: d/dx (nx) = nx ln(n)
Hence, proved.
Higher-Order Derivatives of n^x
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 nx.
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 nx, 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:
When n < 0, the derivative is undefined for non-integer values of x because nx may not be real. When n = 1, the derivative of nx = 0, since 1x is constant.
Common Mistakes and How to Avoid Them in Derivatives of n^x
Students frequently make mistakes when differentiating nx. 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 solving using the chain rule. 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 the Undefined Points of nx
They might not remember that nx is undefined for negative bases n and non-integer exponents x. Keep in mind that you should consider the domain of the function that you differentiate. It will help you understand that the function is not continuous at such certain points.
Mistake 3
Incorrect use of the Chain Rule
While differentiating functions such as n(2x), students misapply the chain rule.
For example: Incorrect differentiation: d/dx (n(2x)) = 2n(2x) ln(n).
Correct application: d/dx (n(2x)) = n(2x) · d/dx (2x) ln(n) = 2n(2x) ln(n). To avoid this mistake, write the chain rule without errors. Always check for errors in the calculation and ensure it is properly simplified.
Mistake 4
Not writing Constants and Coefficients
There is a common mistake that students at times forget to multiply the constants placed before nx.
For example, they incorrectly write d/dx (5nx) = nx ln(n).
Students should check the constants in the terms and ensure they are multiplied properly.
For e.g., the correct equation is d/dx (5nx) = 5nx ln(n).
Mistake 5
Not Applying the Chain Rule
Students often forget to use the chain rule. This happens when the derivative of the inner function is not considered.
For example: Incorrect: d/dx (n(2x)) = n(2x) ln(n).
To fix this error, students should divide the functions into inner and outer parts. Then, make sure that each function is differentiated.
For example, d/dx (n(2x)) = 2n(2x) ln(n).
Examples Using the Derivative of n^x
Problem 1
Calculate the derivative of (n^x·ln(n))
Here, we have f(x) = nx·ln(n).
Using the product rule, f'(x) = u′v + uv′ In the given equation, u = nx and v = ln(n).
Let’s differentiate each term, u′ = d/dx (nx) = nx ln(n) v′ = d/dx (ln(n)) = 0
Substituting into the given equation, f'(x) = (nx ln(n))·0 + (nx)·(ln(n))
Let’s simplify terms to get the final answer, f'(x) = nx ln(n)
Thus, the derivative of the specified function is nx ln(n).
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
An investment grows at a rate represented by the function y = n^x where y represents the value over time x. If x = 2 years, measure the growth rate of the investment.
We have y = nx (growth function)...(1)
Now, we will differentiate the equation (1)
Take the derivative of nx: dy/dx = nx ln(n)
Given x = 2 (substitute this into the derivative) dy/dx = n2 ln(n)
Hence, we get the growth rate of the investment at x = 2 years as n2 ln(n).
Explanation
We find the growth rate of the investment at x = 2 years as n2 ln(n), which means that at a given point, the value of the investment would increase at this rate.
Problem 3
Derive the second derivative of the function y = n^x.
The first step is to find the first derivative, dy/dx = nx ln(n)...(1)
Now we will differentiate equation (1) to get the second derivative: d²y/dx² = d/dx [nx ln(n)]
Using the product rule, d²y/dx² = nx ln(n)·ln(n) + nx ln(n) d²y/dx² = nx (ln(n))² + nx ln(n)
Therefore, the second derivative of the function y = nx is nx (ln(n))² + nx ln(n).
Explanation
We use the step-by-step process, where we start with the first derivative. Using the product rule, we differentiate nx ln(n). We then substitute and simplify the terms to find the final answer.
Problem 4
Prove: d/dx ((n^x)²) = 2n^(2x) ln(n).
Let’s start using the chain rule: Consider y = (nx)²
To differentiate, we use the chain rule: dy/dx = 2(nx)·d/dx [nx]
Since the derivative of nx is nx ln(n), dy/dx = 2(nx)·nx ln(n) dy/dx = 2n(2x) ln(n)
Hence proved.
Explanation
In this step-by-step process, we used the chain rule to differentiate the equation. Then, we replace nx with its derivative. As a final step, we simplify the expression to derive the equation.
Problem 5
Solve: d/dx (n^x/x)
To differentiate the function, we use the quotient rule: d/dx (nx/x) = (d/dx (nx)·x - nx·d/dx(x))/x²
We will substitute d/dx (nx) = nx ln(n) and d/dx (x) = 1 (x nx ln(n) - nx·1) / x² = (x nx ln(n) - nx) / x²
Therefore, d/dx (nx/x) = (x nx ln(n) - nx) / 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 n^x
1.Find the derivative of n^x.
2.Can we use the derivative of n^x in real life?
3.Is it possible to take the derivative of n^x at the point where n < 0?
4.What rule is used to differentiate n^x/x?
5.Are the derivatives of n^x and x^n the same?
6.Can we find the derivative of n^x formula?
Important Glossaries for the Derivative of n^x
- 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 a constant base is raised to a variable exponent, written as nx.
- Natural Logarithm: The logarithm to the base e, where e is an irrational constant approximately equal to 2.71828, denoted as ln(n).
- Chain Rule: A rule for differentiating compositions of functions.
- Quotient Rule: A rule for differentiating functions that are divided by one another.
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.
