Derivative of Negative x

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

The key concepts are mentioned below:

Linear Function: A function of the form y = mx + b, where m and b are constants.

Constant Rule: The derivative of a constant is zero.

Negative Slope: A line with a negative slope descends from left to right.

The derivative of -x can be denoted as d/dx (-x) or (-x)'. The formula we use to differentiate -x is: d/dx (-x) = -1 (or) (-x)' = -1 The formula applies to all x, as the derivative of a linear function is constant.

We can derive the derivative of -x using proofs. To show this, we will use basic differentiation rules. There are a few straightforward methods we use to prove this, such as:

By First Principle

The derivative of -x can be proved using the First Principle, which expresses the derivative as the limit of the difference quotient.

To find the derivative of -x using the first principle, we will consider f(x) = -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) = -x, we write f(x + h) = -(x + h).

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

Using Basic Differentiation Rules

To prove the differentiation of -x using basic rules, consider the linear function y = -x.

The derivative of a constant times a function is the constant times the derivative of the function.

Thus, d/dx (-1 * x) = -1 * d/dx (x) The derivative of x with respect to x is 1. Therefore, d/dx (-x) = -1 * 1 = -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.

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).

For the nth Derivative of -x, we generally use fⁿ(x) for the nth derivative of a function f(x), but for a linear function like -x, all higher-order derivatives beyond the first are zero.

For any real number x, the derivative of -x is always -1. The derivative remains constant at -1 regardless of the value of x.

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

• Linear Function: A function that forms a straight line, typically represented as y = mx + b.

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

• Product Rule: A differentiation rule used for functions that are products of two functions.

• Higher-Order Derivative: Derivatives beyond the first, indicating further rates of change of a function.