Derivative of Straight Line

We now understand the derivative of a straight line. It is commonly represented as d/dx (ax + b) or (ax + b)', and its value is a. The function ax + b has a clearly defined derivative, indicating it is differentiable everywhere.

The key concepts are mentioned below:

Linear Function: (ax + b), where a and b are constants.

Slope: The coefficient a represents the slope of the line.

Constant Function: The derivative of a constant b is zero.

The derivative of a straight line can be denoted as d/dx (ax + b) or (ax + b)'.

The formula to differentiate a straight line is: d/dx (ax + b) = a

The formula applies to all real numbers x.

We can derive the derivative of a straight line using proofs. To show this, we will use the definition of a derivative along with basic differentiation rules.

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

• By First Principle
   
• Using Constant Rule

We will now demonstrate that the differentiation of ax + b results in a using the above-mentioned methods:

By First Principle

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

Using Constant Rule

To prove the differentiation of ax + b using the constant rule, We use the formula: d/dx (ax + b) = d/dx (ax) + d/dx (b) = a + 0 (since the derivative of a constant is zero) = a Therefore, the derivative of ax + b is a.

When a function is differentiated several times, the derivatives obtained are referred to as higher-order derivatives. For a straight line, higher-order derivatives are straightforward. The first derivative of a linear function is the slope, which is constant. The second derivative of a linear function is zero, as the rate of change of the slope is nonexistent.

The first derivative of a function, denoted as f′(x), indicates the slope of the line. The second derivative, f′′(x), is the derivative of the first derivative and is zero for a linear function. Similarly, all higher-order derivatives are zero.

When the line is horizontal (a = 0), the derivative is zero because there is no change in y with respect to x.

When the line is vertical (x = constant), the derivative is undefined because the rise over run is infinite.

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

• Linear Function: A function of the form ax + b, where a and b are constants.

• Slope: The coefficient of x in a linear function, representing its rate of change.

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

• Higher-Order Derivatives: Derivatives beyond the first derivative, which are zero for linear functions.