Derivative of 1 | Differentiation of 1

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.

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(xⁿ) = 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 = x0
Differentiate both sides w.r.t. x	d/dx(1) = d/dx(x0)
Apply power rule of derivative	d/dx(1) = d/dx(x0) = 0 x0-1 = 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/x2
Derivative of sin 3x:	3cos 3x
Derivative of esin x :	cos x esin 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=x⁰. So by power rule, d/dx(1) = d/dx(x⁰) = 0 x^0-1 = 0.
• The derivative of any whole number is zero.

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