Derivative of sin(x²)

We now understand the derivative of sin(x²).

It is commonly represented as d/dx (sin(x²)) or (sin(x²))', and its value is 2x cos(x²).

The function sin(x²) has a clearly defined derivative, indicating it is differentiable for all real numbers.

The key concepts are mentioned below:

Sine Function: (sin(x²) is the sine of the square of x).

Chain Rule: Rule for differentiating sin(x²) due to the composition of functions.

Cosine Function: cos(x) is the cosine of x.

The derivative of sin(x²) can be denoted as d/dx (sin(x²)) or (sin(x²))'. The formula we use to differentiate sin(x²) is: d/dx (sin(x²)) = 2x cos(x²) The formula applies to all real numbers x.

We can derive the derivative of sin(x²) using proofs. To show this, we will use 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

Using Product Rule

We will now demonstrate that the differentiation of sin(x²) results in 2x cos(x²) using the above-mentioned methods:

Using Chain Rule

To prove the differentiation of sin(x²) using the chain rule, We use the formula: y = sin(u), where u = x² Then dy/du = cos(u) and du/dx = 2x

Using the chain rule: dy/dx = dy/du * du/dx dy/dx = cos(x²) * 2x

Thus, d/dx (sin(x²)) = 2x cos(x²).

Using First Principle The derivative of sin(x²) can be proved using the First Principle, which expresses the derivative as the limit of the difference quotient.

To find the derivative of sin(x²) using the first principle, we will consider f(x) = sin(x²).

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(x²), we write f(x + h) = sin((x + h)²).

Substituting these into equation (1), f'(x) = limₕ→₀ [sin((x + h)²) - sin(x²)] / h

Using trigonometric identities and limit properties, and simplifying, f'(x) = 2x cos(x²).

Using Product Rule

The product rule is not directly applicable to sin(x²) as it is not a product of functions. But for functions involving multiplication of derivatives, the product rule can be useful.

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(x²).

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(x²), 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 0, the derivative of sin(x²) = 2x cos(x²), which is 0, because the term 2x evaluates to 0. For any other points, the derivative is well-defined and can be calculated using the formula.

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

• Chain Rule: A fundamental rule used in calculus to differentiate composite functions.

• Sine Function: A trigonometric function that represents the y-coordinate of a point on the unit circle.

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

• First Derivative: The initial result of a function's differentiation, which gives the rate of change of the function.