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.