Derivative of 1-x

We now understand the derivative of 1-x. It is commonly represented as d/dx (1-x) or (1-x)', and its value is -1. The function 1-x has a clearly defined derivative, indicating it is differentiable within its domain.

The key concepts are mentioned below:

Linear Function: (1-x is a linear function).

Constant Rule: The derivative of a constant is 0.

Power Rule: Used for differentiating terms of the form xⁿ.

The derivative of 1-x can be denoted as d/dx (1-x) or (1-x)'.

The formula we use to differentiate 1-x is: d/dx (1-x) = -1 (or) (1-x)' = -1

The formula applies to all x in the real number domain.

We can derive the derivative of 1-x using proofs. To show this, we will use basic differentiation rules.

There are several methods we use to prove this, such as:

• By First Principle
   
• Using Power Rule

We will now demonstrate that the differentiation of 1-x results in -1 using the above-mentioned methods:

By First Principle

The derivative of 1-x can be proved using the First Principle, which expresses the derivative as the limit of the difference quotient. To find the derivative of 1-x using the first principle, we will consider f(x) = 1-x. Its derivative can be expressed as the following limit. f'(x) = limₕ→₀ [f(x + h) - f(x)] / h … (1) Given that f(x) = 1-x, we write f(x + h) = 1-(x + h). Substituting these into equation (1), f'(x) = limₕ→₀ [1-(x + h) - (1-x)] / h = limₕ→₀ [-h] / h = limₕ→₀ -1 f'(x) = -1. Hence, proved.

Using Power Rule

To prove the differentiation of 1-x using the power rule, We use the formula: 1-x = 1 - x¹ The derivative of a constant is 0, and the derivative of x¹ is 1. Therefore, d/dx (1-x) = 0 - 1 = -1.

When a function is differentiated several times, the derivatives obtained are referred to as higher-order derivatives. Higher-order derivatives can be simple for linear functions. To understand them better, think of a constant velocity where the speed remains the same, and the acceleration (second derivative) is zero. Higher-order derivatives make it easier to understand functions like 1-x.

For the first derivative of a function, we write f′(x), which indicates how the function changes or its slope at a certain point. The second derivative of a linear function like 1-x is 0, denoted using f′′(x).

For the nth Derivative of 1-x, we generally use fⁿ(x) for the nth derivative of a function f(x), which tells us the change in the rate of change. In this case, all higher-order derivatives are 0.

For any x, the derivative of 1-x is always -1, as it is a constant slope.

For any change in x, the change in the y-value of the function 1-x is constant.

• Derivative: The derivative of a function indicates how the given function changes in response to a slight change in x.

• Linear Function: A function of the form ax+b, where the graph is a straight line.

• Constant Rule: A rule stating that the derivative of a constant is 0.

• Power Rule: A basic rule for differentiating functions of the form xⁿ.

• Chain Rule: A rule used for differentiating compositions of functions.