Derivative of sin(πx)

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

The key concepts are mentioned below:

Sine Function: sin(πx).

Chain Rule: Rule for differentiating sin(πx) (since it involves a composite function).

Cosine Function: cos(x).

The derivative of sin(πx) can be denoted as d/dx (sin(πx)) or (sin(πx))'. The formula we use to differentiate sin(πx) is: d/dx (sin(πx)) = πcos(πx)

The formula applies to all x.

We can derive the derivative of sin(πx) using proofs. To show this, we will use the trigonometric identities along with the rules of differentiation.

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

• By First Principle
   
• Using Chain Rule

We will now demonstrate that the differentiation of sin(πx) results in πcos(πx) using the above-mentioned methods:

By First Principle

The derivative of sin(πx) can be proved using the First Principle, which expresses the derivative as the limit of the difference quotient. To find the derivative of sin(πx) using the first principle, we will consider f(x) = sin(πx). Its derivative can be expressed as the following limit. f'(x) = limₕ→₀ [f(x + h) - f(x)] / h … (1) Given that f(x) = sin(πx), we write f(x + h) = sin(π(x + h)).

Substituting these into equation (1), f'(x) = limₕ→₀ [sin(π(x + h)) - sin(πx)] / h = limₕ→₀ [sin(πx + πh) - sin(πx)] / h Using the formula sin A - sin B = 2 cos((A+B)/2) sin((A-B)/2), f'(x) = limₕ→₀ [2 cos(πx + πh/2) sin(πh/2)] / h = limₕ→₀ [2 cos(πx + πh/2) (sin(πh/2)/(πh/2)) (π/2)] Using limit formulas, limₕ→₀ (sin(πh/2)/(πh/2)) = 1. f'(x) = π cos(πx) Hence, proved.

Using Chain Rule

To prove the differentiation of sin(πx) using the chain rule, We use the formula: Let u(x) = πx, then sin(πx) = sin(u(x)) The derivative of sin(u) is cos(u) times the derivative of u, d/dx (sin(πx)) = cos(πx) * d/dx (πx) = cos(πx) * π = πcos(πx) Thus, the derivative of sin(πx) is πcos(πx).

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 sin(πx).

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 sin(πx), we generally use fⁿ(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).

When x is such that πx is an odd multiple of π/2, the derivative is zero because cos(πx) is zero at these points.

When x is 0, the derivative of sin(πx) = πcos(0), which is π.

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

• Sine Function: A trigonometric function written as sin(x), representing the y-coordinate of a point on the unit circle.

• Cosine Function: A trigonometric function representing the x-coordinate of a point on the unit circle.

• Chain Rule: A rule in calculus for differentiating compositions of functions.

• Quotient Rule: A formula to find the derivative of a quotient of two functions.