Derivative of pix/2

We now understand the derivative of pix/2. It is commonly represented as d/dx (pix/2) or (pix/2)', and its value is pi/2. The function pix/2 has a clearly defined derivative, indicating it is differentiable within its domain.

The key concepts are mentioned below:

Multiplication Constant: When multiplying a constant with a variable.

Constant Rule: The derivative of a constant.

The derivative of pix/2 can be denoted as d/dx (pix/2) or (pix/2)'.

The formula we use to differentiate pix/2 is: d/dx (pix/2) = pi/2 (Typically, when differentiating a constant multiplied by x, the result is the constant itself.)

The formula applies to all x.

We can derive the derivative of pix/2 using proofs. To show this, we will use basic differentiation rules.

There are several methods to prove this, such as:

1. By First Principle
2. Using Constant Rule

We will now demonstrate that the differentiation of pix/2 results in pi/2 using the above-mentioned methods:

By First Principle

The derivative of pix/2 can be proved using the First Principle, which expresses the derivative as the limit of the difference quotient. To find the derivative of pix/2 using the first principle, we will consider f(x) = pix/2.

Its derivative can be expressed as the following limit. f'(x) = limₕ→₀ [f(x + h) - f(x)] / h … (1) Given that f(x) = pix/2, we write f(x + h) = pi(x + h)/2.

Substituting these into equation (1), f'(x) = limₕ→₀ [pi(x + h)/2 - pix/2] / h = limₕ→₀ [pix/2 + pih/2 - pix/2] / h = limₕ→₀ [pih/2] / h = limₕ→₀ (pi/2) Using limit laws, f'(x) = pi/2.

Hence, proved.

Using Constant Rule

To prove the differentiation of pix/2 using the constant rule, We use the formula: If y = kx, then dy/dx = k. For f(x) = pix/2, dy/dx = pi/2.

Hence, proved.

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 pix/2.

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).

Since the derivative of pix/2 is a constant, all higher-order derivatives are zero.

Since pix/2 is a linear function, its derivative is constant and does not depend on x. At any point x, the derivative of pix/2 is pi/2.

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

• Constant Rule: A basic differentiation rule stating that the derivative of a constant multiplied by x is the constant itself.

• First Derivative: The initial result of differentiating a function, indicating the rate of change of the function.

• Linear Function: A function of the form f(x) = ax + b, where the first derivative is constant.

• Higher-Order Derivative: The result of repeatedly differentiating a function, where for constants, these are zero after the first derivative.