# The Derivative of a Constant (With Examples)

*Ray is a Licensed Engineer in the Philippines. He loves to write any topic about mathematics and civil engineering.*

The derivative of a constant is always zero**. **The Constant Rule states that if f(x) = c, then f’(c) = 0 considering c is a constant. In Leibniz notation, we write this differentiation rule as follows:

**d/dx (c) = 0**

A constant function is a function, whereas its y does not change for variable x. In layman's terms, constant functions are functions that do not move. They are principally numbers. Consider constants as having a variable raised to the power zero. For instance, a constant number 5 can be 5x0, and its derivative is still zero.

The derivative of a constant function is one of the most basic and most straightforward differentiation rules that students must know. It is a rule of differentiation derived from the power rule that serves as a shortcut to finding the derivative of any constant function and bypassing solving limits. The rule for differentiating constant functions and equations is called the Constant Rule.

The Constant Rule is a differentiation rule that deals with constant functions or equations, even if it is a π, Euler's number, square root functions, and more. In graphing a constant function, the result is a horizontal line. A horizontal line imposes a constant slope, which means there is no rate of change and slope. It suggests that for any given point of a constant function, the slope is always zero.

## Why Is the Derivative of a Constant Zero?

Ever wondered why the derivative of a constant is 0?

We know that dy/dx is a derivative function, and it also means that the values of y are changing for the values of x. Hence, y is dependent on the values of x. Derivative means the limit of the change ratio in a function to the corresponding change in its independent variable as the last change approaches zero.

A constant remains constant irrespective of any change to any variable in the function. A constant is always a constant, and it is independent of any other values existing in a particular equation.

The derivative of a constant comes from the definition of a derivative.

f′(x) = lim h→0 [f(x+h)−f(x)] / h

f′(x) = lim h→0 (c−c) /h

f′(x) = lim h→0 0

f′(x) = 0

To further illustrate that the derivative of a constant is zero, let us plot the constant on the y-axis of our graph. It will be a straight horizontal line as the constant value does not change with the change in the value of x on the x-axis. The graph of a constant function f(x) = c is the horizontal line y=c which has slope = 0. So, the first derivative f' (x) is equal to 0.

## Example 1: Derivative of a Constant Equation

What is the derivative of y = 4?

**Answer**

The first derivative of y = 4 is y’ = 0.

## Example 2: Derivative of a Constant Equation F(X)

Find the derivative of the constant function f(x) = 10.

**Answer**

The first derivative of the constant function f(x) = 10 is f’(x) = 0.

## Example 3: Derivative of a Constant Function T(X)

What is the derivative of the constant function t(x) = 1?

**Answer**

The first derivative of the constant function t(x) = 1 is t’(x) = 1.

## Example 4: Derivative of a Constant Function G(X)

Find the derivative of the constant function g(x) = 999.

**Answer**

The first derivative of the constant function g(x) = 999 is still g’(x) = 0.

## Example 5: Derivative of Zero

Find the derivative of 0.

**Answer**

The derivative of 0 is always 0. This example still falls under the derivative of a constant.

## Example 6: Derivative of Pi

What is the derivative of π?

**Answer**

The value of π is 3.14159. Still a constant, so the derivative of π is zero.

## Example 7: Derivative of a Fraction with a Constant Pi

Find the derivative of the function (3π + 5) / 10.

**Answer**

The given function is a complex constant function. Therefore, its first derivative is still 0.

## Example 8: Derivative of the Euler's Number "e"

What is the derivative of the function √(10) / (e−1)?

**Answer**

The exponential** **"e" is a numerical constant that is equal to 2.71828. Technically, the function given is still constant. Hence, the first derivative of the constant function is zero.

## Example 9: Derivative of a Fraction

What is the derivative of the fraction 4 / 8?

**Answer**

The derivative of 4 / 8 is 0.

## Example 10: Derivative of a Negative Constant

What is the derivative of the function f(x) = -1099?

**Answer**

The derivative of the function f(x) = -1099 is 0.

## Example 11: Derivative of a Constant to a Power

Find the derivative of *e*^{x}.

**Answer**

Note that *e *is a constant and has a numerical value. The given function is a constant function raised to the power of x. According to the derivative rules, the derivative of *e*^{x} is the same as its function. The slope of the function *e*^{x} is constant, wherein that for every x-value, the slope is equal to every y-value. Therefore, the derivative of *e*^{x} is 0.

## Example 12: Derivative of a Constant Raised to the X Power

What is the derivative of 2^{x}?

**Answer**

Rewrite 2 to a format that contains an Euler number *e.*

2^{x} = (*e *ln (2))^{x} ln(2)

2_{x} = 2^{x} ln (2)

Therefore, the derivative of 2^{x} is 2^{x} ln(2).

## Example 13: Derivative of a Square Root Function

Find the derivative of y = √81.

**Answer**

The given equation is a square root function √81. Remember that a square root is a number multiplied by it to get the resulting number. In this case, √81 is 9. The resulting number 9 is called the square of a square root.

Following the Constant Rule, the derivative of an integer is zero. Therefore, f' (√81) is equal to 0.

## Example 14: Derivative of a Trigonometric Function

Extract the derivative of the trigonometric equation y = sin (75°).

**Answer**

The trigonometric equation sin (75°) is a form of sin (x) where x is any degree or radian angle measure. If to get the numerical value of sin (75°), the resulting value is 0.969. Given that sin (75°) is 0.969. Therefore, its derivative is zero.

## Example 15: Derivative of a Summation

Given the summation ∑_{x=1}^{10} (x^{2})

**Answer**

The given summation has a numerical value, which is 385. Thus, the given summation equation is a constant. Since it is a constant, y' = 0.

## Explore Other Calculus Articles

- Solving Related Rates Problems in Calculus

Learn to solve different kinds of related rates problems in Calculus. This article is a full guide that shows the step-by-step procedure of solving problems involving related/associated rates. - Limit Laws and Evaluating Limits

This article will help you learn to evaluate limits by solving various problems in Calculus that require applying the limit laws.

*This content is accurate and true to the best of the author’s knowledge and is not meant to substitute for formal and individualized advice from a qualified professional.*

**© 2020 Ray**