Derivative of -x

The derivative of -x is straightforward to understand. It is commonly represented as d/dx (-x) or (-x)', and its value is -1.

This indicates that for every unit increase in x, the function value decreases by 1. This linear function has a constant slope and is differentiable across its entire domain.

Key concepts include: Linear Function: (-x) is a linear function with a constant slope.

Constant Rule: The derivative of a constant multiplied by a function is the constant multiplied by the derivative of the function.

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

This formula is valid for all x in the real number line.

We can prove the derivative of -x using different approaches.

The most straightforward method is to use the basic rules of differentiation.

Here are the methods we can use: By the Constant Rule The derivative of -x can be derived using the constant rule, which states that the derivative of a constant multiplied by a function is the constant times the derivative of the function.

Let f(x) = -x, which can be rewritten as f(x) = -1 * x. The derivative, f'(x), is -1 * d/dx (x).

Since the derivative of x is 1, we have: f'(x) = -1 * 1 = -1. Hence, the derivative of -x is -1. Using the First Principle We can also prove the derivative of -x using the first principle, which defines the derivative as the limit of the difference quotient.

Consider f(x) = -x. The derivative is expressed as the following limit: f'(x) = limₕ→₀ [f(x + h) - f(x)] / h = limₕ→₀ [-(x + h) + x] / h = limₕ→₀ [-h] / h = limₕ→₀ -1 Thus, f'(x) = -1. Hence, proved by the first principle.

When a function is differentiated several times, the derivatives obtained are referred to as higher-order derivatives.

For the function -x, the process is simple due to its linear nature.

The first derivative of -x is -1, which indicates the rate of change is constant.

The second derivative, derived from the first derivative, is 0, indicating no change in the slope.

Similarly, all higher-order derivatives of -x are 0.

Since -x is a linear function with constant slope, there are no special cases of discontinuity or undefined points.

The function is continuous and differentiable across the entire 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 function of the form y = mx + b, where m and b are constants; its derivative is constant.

• Constant Rule: In differentiation, the derivative of a constant times a function is the constant times the derivative of the function.

• Higher-Order Derivatives: Successive derivatives of a function, indicating rates of change of different orders.

• Rate of Change: The derivative represents the rate at which a function value changes with respect to changes in the independent variable.