Derivative of x-1

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

The key concepts are mentioned below: Linear Function: Functions of the form ax + b.

Constant Rule: The derivative of any constant is zero.

Power Rule: For functions of the form x^n, the derivative is n*x^(n-1).

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

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

The formula applies to all x in the domain of real numbers.

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

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

1. By First Principle Using Power Rule
2. Using Sum Rule

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

By First Principle

The derivative of x-1 can be proved using the First Principle, which expresses the derivative as the limit of the difference quotient. To find the derivative of x-1 using the first principle, we will consider f(x) = x-1.

Its derivative can be expressed as the following limit. f'(x) = limₕ→₀ [f(x + h) - f(x)] / h

Given that f(x) = x-1, we write f(x + h) = (x + h) - 1.

Substituting these into the equation, f'(x) = limₕ→₀ [(x + h - 1) - (x - 1)] / h = limₕ→₀ [x + h - 1 - x + 1] / h = limₕ→₀ h / h = limₕ→₀ 1 Hence, f'(x) = 1.

Using Power Rule

To prove the differentiation of x-1 using the power rule, We use the formula: d/dx (x^n) = n*x^(n-1) For f(x) = x, we have n = 1. d/dx (x) = 1*x^(1-1) = 1

Using Sum Rule We will now prove the derivative of x-1 using the sum rule.

The step-by-step process is demonstrated below: The sum rule states that the derivative of a sum is the sum of the derivatives.

Given f(x) = x - 1, we have two terms: x and -1.

The derivative of x is 1 (using the power rule), and the derivative of -1 is 0 (using the constant rule).

Thus, d/dx (x - 1) = 1 + 0 = 1

When a function is differentiated several times, the derivatives obtained are referred to as higher-order derivatives. Higher-order derivatives can be a little tricky.

To understand them better, think of a car where the speed changes (first derivative) and the rate at which the speed changes (second derivative) also changes. Higher-order derivatives make it easier to understand functions like x-1.

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 is derived from the first derivative, which is denoted using f′′ (x). Similarly, the third derivative, f′′′(x), is the result of the second derivative, and this pattern continues.

For the nth Derivative of x-1, we generally use f n(x) for the nth derivative of a function f(x), which tells us the change in the rate of change. (continuing for higher-order derivatives).

For the function x-1, the derivative is always 1, regardless of the value of x. The function is continuous and differentiable at all points on the real number line.

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

• Linear Function: A linear function is a function of the form ax+b.

• Constant Rule: The derivative of any constant is zero.

• Power Rule: For functions of the form x^n, the derivative is n*x^(n-1).

• Higher-order Derivative: The derivative of a derivative, indicating the rate of change at multiple levels. ```