Derivative of Sin(1/x)

We now understand the derivative of sin(1/x). It is commonly represented as d/dx [sin(1/x)] or [sin(1/x)]', and its value involves the chain rule and is given by cos(1/x) * (-1/x²). The function sin(1/x) has a clearly defined derivative, indicating it is differentiable within its domain.

The key concepts are mentioned below:

Sine Function: sin(θ) is a trigonometric function.

Chain Rule: A rule for differentiating compositions of functions, essential for sin(1/x).

Cosine Function: cos(θ) is another basic trigonometric function.

The derivative of sin(1/x) can be denoted as d/dx [sin(1/x)] or [sin(1/x)]'. The formula we use to differentiate sin(1/x) is: d/dx [sin(1/x)] = cos(1/x) * (-1/x²) The formula applies to all x where x ≠ 0, as the function is undefined at x = 0.

We can derive the derivative of sin(1/x) 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:

1. By First Principle
2. Using Chain Rule

We will now demonstrate that the differentiation of sin(1/x) results in cos(1/x) * (-1/x²) using the above-mentioned methods:

Using Chain Rule

To prove the differentiation of sin(1/x) using the chain rule, Consider y = sin(1/x). Let u = 1/x, then y = sin(u).

By the chain rule: dy/dx = dy/du * du/dx dy/du = cos(u) = cos(1/x) du/dx = -1/x² Thus, dy/dx = cos(1/x) * (-1/x²)

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 help us understand functions like sin(1/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). Similarly, the third derivative, f′′′(x), is the result of the second derivative, and this pattern continues.

For the nth Derivative of sin(1/x), 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 approaches 0, the function sin(1/x) oscillates and does not have a well-defined limit, leading to discontinuity. When x is large, the derivative cos(1/x) * (-1/x²) approaches 0, indicating the function becomes flatter.

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

• Chain Rule: A rule for finding the derivative of a composition of functions.

• Sine Function: A primary trigonometric function written as sin(θ).

• Cosine Function: A trigonometric function represented as cos(θ), the derivative of the sine function.

• Quotient Rule: A technique used to differentiate functions that are divided by each other.