Derivative of sin²x

We now understand the derivative of sin²x. It is commonly represented as d/dx (sin²x) or (sin²x)', and its value is 2sin(x)cos(x) or sin(2x). The function sin²x has a clearly defined derivative, indicating it is differentiable within its domain. The key concepts are mentioned below: - Sine Function: (sin(x) is a trigonometric function). - Chain Rule: Used for differentiating composite functions like sin²(x). - Double Angle Formula: sin(2x) = 2sin(x)cos(x).

The derivative of sin²x can be denoted as d/dx (sin²x) or (sin²x)'. The formula we use to differentiate sin²x is: d/dx (sin²x) = 2sin(x)cos(x) or (sin²x)' = sin(2x) The formula applies to all x where the sine function is defined.

We can derive the derivative of sin²x 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 Chain Rule - Using Product Rule We will now demonstrate that the differentiation of sin²x results in 2sin(x)cos(x) using the above-mentioned methods: Using Chain Rule To prove the differentiation of sin²x using the chain rule, Consider y = (sin(x))² Let u = sin(x), then y = u² By the chain rule: dy/dx = 2u du/dx Since du/dx = cos(x), dy/dx = 2sin(x)cos(x) Using the double angle formula, sin(2x) = 2sin(x)cos(x), we have: dy/dx = sin(2x) Hence, proved. Using Product Rule We will now prove the derivative of sin²x using the product rule. The step-by-step process is demonstrated below: Here, we use the formula, sin²x = sin(x)·sin(x) Let u = sin(x) and v = sin(x) Using the product rule formula: d/dx [u·v] = u'·v + u·v' u' = d/dx (sin x) = cos x v' = d/dx (sin x) = cos x d/dx (sin²x) = cos(x)·sin(x) + sin(x)·cos(x) = 2sin(x)cos(x) Thus, d/dx (sin²x) = sin(2x) 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 sin²(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 sin²(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.

- When x is 0, the derivative of sin²x = sin(2·0) = 0. - When x is π/4, the derivative of sin²x = sin(π/2) = 1.

Derivative: The derivative of a function indicates how the given function changes in response to a slight change in x. Sine Function: The sine function is one of the primary trigonometric functions, written as sin(x). Chain Rule: A differentiation rule used for composite functions. Double Angle Formula: A trigonometric identity used to express trigonometric functions of double angles, like sin(2x) = 2sin(x)cos(x). Quotient Rule: A rule for differentiating functions that are divided by one another.