The derivative of 1 is zero. Note that 1 can be treated as a constant function. In this blog post, we will learn how to find the derivative of the constant 1 using different methods.

Table of Contents

## Derivative of 1 Formula

The natural number 1 is a constant. As the derivative of a constant is 0, we deduce that the derivative of one is zero.

Thus, the formula for the derivative of 1 is as follows:

d/dx(1) = 0 **or** (1)’ = 0.

## Derivative of 1 by Power Rule

Recall the power rule of derivatives: The derivative of x to the power n is given by the formula:

d/dx(x^{n}) = nx^{n-1}.

To find the derivative of 1, we will follow the below steps described in the table below:

Write 1 as the power of x | 1 = x^{0} |

Differentiate both sides w.r.t. x | d/dx(1) = d/dx(x^{0}) |

Apply power rule of derivative | d/dx(1) = d/dx(x^{0}) = 0 x^{0-1} = 0 |

Conclusion: | The derivative of 1 is equal to 0 |

## Derivative of 1 by First Principle

The derivative of a function f(x) by the first principle is determined by the following limit:

$\dfrac{d}{dx}(f(x))$ $=\lim\limits_{h \to 0} \dfrac{f(x+h)-f(x)}{h}$.

**Step 1:** Put f(x) = 1 in the above formula. Note that f(x+h)=1. So the derivative of the constant 1 by the above first principle is

$\dfrac{d}{dx}(1)$ $=\lim\limits_{h \to 0} \dfrac{1-1}{h}$.

**Step 2:** Simplifying the above, we obtain that

$\dfrac{d}{dx}(1)$ $=\lim\limits_{h \to 0} \dfrac{0}{h}$

= $\lim\limits_{h \to 0} \dfrac{0}{h}$

= $\lim\limits_{h \to 0} 0$

= 0

**Final Answer:** Thus, we conclude that the derivative of 1 is zero, and this is obtained by the first principle of derivatives.

**Video Solution of Derivative of 1 by First Principle:**

**Also Read:**

Derivative of 1/x: | -1/x^{2} |

Derivative of sin 3x: | 3cos 3x |

Derivative of e :^{sin x} | cos x e^{sin x} |

Derivative of log 3x: | 1/x |

### Important Notes on Derivative of 1

- 1 is a constant, so its derivative is zero.
- We can write 1=x
^{0}. So by power rule, d/dx(1) = d/dx(x^{0}) = 0 x^{0-1}= 0. - The derivative of any whole number is zero.

## FAQs on Derivative of 1

**Q1: What is the derivative of 1?**

Answer: The derivative of 1 is zero.

**Q2: What is the derivative of 2?**

Answer: The derivative of 2 is zero.

**Q3: What is the derivative of a number?**

Answer: As any number is a constant and we know that the derivative of a constant is zero, the derivative of any number will be 0.

This article is written by Dr. T. Mandal, Ph.D in Mathematics. On Mathstoon.com you will find Maths from very basic level to advanced level. Thanks for visiting.