Derivative of u^x

Learn what is the derivative of u^x with formula. Also understand how to verify the derivative of an algebraic function u^x by using first principle.

Alan Walker-

Published on 2023-06-28

Introduction to the derivative of u^x

Derivatives have a wide range of applications in almost every field of engineering and science. The u^x derivative is equal to the product of u^x and log n which can be calculated by following the rules of differentiation. Or, we can directly find the derivative formula of u^x by applying the first principle of differentiation. In this article, you will learn what the u^x derivative formula is and how to calculate the derivatives of u^x by using different approaches.

What is the derivative of u^x?

The derivative of u^x is equal to u^x log n. It measures the rate of change of the exponential function u^x. It is denoted by d/dx(u^x) which is a fundamental concept in calculus.

Knowing the formula for derivatives and understanding how to use it can be used in solving problems related to velocity, acceleration, and optimization. 

Derivative of u^x formula

The formula for derivative of f(x)=u^x is equal to the u^x log n, that is;

$f'(x) = \frac{d}{dx} (u^x) = u^x log u$

It is calculated by using the logarithmic differentiation.

How do you differentiate u^x?

There are multiple ways to prove the differentiation of u x. These are;

  1. First Principle
  2. Logarithmic differentiation
  3. Quotient Rule

Each method provides a different way to compute the u^x differentiation. By using these methods, we can mathematically prove the formula for finding the differential of u^x.

Derivative of u power x by first principle

According to the first principle of derivative, the u^x derivative is equal to u^x log n. The derivative of a function by first principle refers to finding a general expression for the slope of a curve by using algebra. It is also known as the delta method. The derivative is a measure of the instantaneous rate of change, which is equal to,

$f’(x) = \lim_{h\to 0}\frac{f(x + h) - f(x)}{h}$

This formula allows us to determine the rate of change of a function at a specific point by using the limit definition of derivative. 

Proof of u^x derivative formula by first principle

To prove the derivative of u^x by using first principle, replace f(x) by u^x or you can replace it by 3^n to find the derivative of 3^n

$f(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}$

$f'(x) = \lim_{h \to 0} \frac{u^{x + h} - u^x}{h}$


$f'(x) = \lim_{h \to 0} \frac{u^x+u^h-u^x}{h}$

$f'(x) = \lim_{h \to 0} \frac{u^x(u^h-1)}{h}$

When h approaches to zero,

$\lim_{h\to 0}\frac{u^h-1}{h}=\log u$


$f'(x) = u^x \log u$

Hence the differentiation of u x is equal to u^x log u. Use our derivative definition calculator to simplify above calculations easily. 

Derivative of u^x by Logarithmic Differentiation

Logarithmic differentiation is a technique of solving derivatives of logarithmic functions. A logarithmic function is the inverse of an exponential function and can be written using a base of 10. It is a method of finding derivatives of complex functions by applying logarithms.

Proof of differentiating of u^x by Logarithmic Differentiation

To differentiate of u^x by using the logarithmic differentiation, we start by assuming that,

$y = u^x$

Taking log on both sides.

$\log y=\log u^x$

$\log y=x \log u$

Now differentiating on the both sides, 

$\frac{d}{dx} (\log y) = \log u\frac{dx}{dx}$

Since the derivative of log x is equal to the reciprocal of x, therefore, 

$\frac{dy}{dx}.\frac{1}{y}= \log u$


$\frac{dy}{dx} = y\log u$

Substituting the value of y, we get

$\frac{dy}{dx} = u^x\log u$

Hence we can derive the derivative of u x by using two derivative rules i.e. delta method and the logarithmic differentiation. 

How to find the derivatives of u^x with a calculator?

The easiest way of differentiating u^x is by using an online calculator. You can use our derivative calculator with steps for this. Here, we provide you a step-by-step way to calculate derivatives by using this tool.

  1. Write the function as u^x in the enter function box. In this step, you need to provide input value as a function that you want to differentiate.
  2. Now, select the variable by which you want to differentiate u^x. Here you have to choose x.
  3. Select how many times you want to calculate the derivatives of u^x. In this step, you can choose 2 to calculate the second derivative, 3 for the third derivative and so on.
  4. Click on the calculate button. After this step, you will get the derivative of x cube within a few seconds.

After completing these steps, you will receive the differential of u x within seconds. Using online tools like derivative calculator can make it much easier and faster to calculate derivatives, especially for complex functions.


In conclusion, the derivative of u^x is u^x log u. The derivative measures the rate of change of a function with respect to its independent variable, and in the case of u^x, the derivative can be calculated using the first principle of derivative and the logarithmic differentiation. 

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