Derivative of b^x

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

Alan Walker-

Published on 2023-06-21

Introduction to the derivative of b^x

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

What is the derivative of b^x?

The derivative of b to the x is equal to b^x log b. It measures the rate of change of the exponential function b^x. It is denoted by d/dx(b^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. 

Fifferentiation of b^x formula

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

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

It is calculated by using the logarithmic differentiation.

How do you differentiate b^x?

There are multiple ways to prove the b^x derivative. These are;

  1. First Principle
  2. Logarithmic differentiation

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

Derivative of b power x by first principle

According to the first principle of derivative, the d/dx b^x is equal to b^x log b. 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 derivative of b to the x formula by first principle

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

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

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


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

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

When h approaches to zero,

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


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

Hence the derivative b^x is equal to b^x log b. You can also use our derivative definition calculator to simplify the above calculations easily. 

b^x derivative 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 b^x by Logarithmic Differentiation

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

$y = b^x$

Taking log on both sides.

$\log y=\log b^x$

$\log y=x \log b$

Now differentiating on the both sides, 

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

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

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


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

Substituting the value of y, we get

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

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

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

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

  1. Write the function as b^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 b^x. Here you have to choose x.
  3. Select how many times you want to calculate the derivative b^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 b x within a few seconds.

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


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

Frequently Asked Questions

What is the derivative of b^x?

The b^x derivative is equal to the product of b^x and log b. It is denoted by d/dx(b^x) and written as d/dx(b^x) = b^x log b, where b is a constant. 

What is the derivative of a to the power x?

The derivative of a to the power x is equal to the product of a^x and log a. It is denoted by d/dx(a^x) and written as d/dx(a^x) = a^x log a, where a is a constant. 

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