Derivative of 2sinxcosx

We now understand the derivative of 2sinxcosx. This function simplifies to sin(2x), which is differentiable within its domain. The derivative of sin(2x) is commonly represented as d/dx (sin(2x)) or (sin(2x))', and its value is 2cos(2x). The key concepts are mentioned below: - Double Angle Formula: 2sinxcosx = sin(2x). - Derivative of Sine: The derivative of sin(x) is cos(x). - Chain Rule: Rule for differentiating composite functions like sin(2x).

The derivative of 2sinxcosx can be denoted as d/dx (2sinxcosx) or (2sinxcosx)'. The formula we use to differentiate 2sinxcosx, which simplifies to sin(2x), is: \[d/dx (sin(2x)) = 2cos(2x)\] The formula applies to all x within the domain of the cosine function.

We can derive the derivative of 2sinxcosx using proofs. To show this, we will use trigonometric identities along with differentiation rules. Several methods can be used to prove this, such as: - Using the Double Angle Identity - Applying the Chain Rule - Using the Product Rule We will demonstrate that the differentiation of 2sinxcosx results in 2cos(2x) using these methods: Using the Double Angle Identity The function 2sinxcosx can be rewritten using the double angle identity: \[2sinxcosx = sin(2x)\] The derivative of sin(2x) is found using the chain rule: \[d/dx (sin(2x)) = 2cos(2x)\] Applying the Chain Rule To differentiate sin(2x) using the chain rule, let: \[u = 2x\] \[d/dx (sin(u)) = cos(u) \cdot du/dx = cos(2x) \cdot 2 = 2cos(2x)\] Using the Product Rule For the product 2sinxcosx, let: \[u = 2sinx \quad \text{and} \quad v = cosx\] \[d/dx (u \cdot v) = u' \cdot v + u \cdot v'\] \[u' = 2cosx, \quad v' = -sinx\] \[d/dx (2sinxcosx) = 2cosx \cdot cosx + 2sinx \cdot (-sinx)\] \[= 2cos^2x - 2sin^2x = 2(cos^2x - sin^2x) = 2cos(2x)\]

When a function is differentiated multiple times, the resulting derivatives are referred to as higher-order derivatives. Higher-order derivatives can be complex. For instance, consider 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 analyze functions like 2sinxcosx. For the first derivative of a function, we write f′(x), indicating the function's rate of change or slope at a point. The second derivative is derived from the first derivative, denoted as f′′(x). Similarly, the third derivative, f′′′(x), is derived from the second derivative, and this pattern continues. For the nth Derivative of 2sinxcosx, we use fⁿ(x) to denote the nth derivative of a function f(x), describing the change in the rate of change.

At x = π/2, the derivative of 2sinxcosx is 2cos(π) = -2. At x = 0, the derivative of 2sinxcosx = 2cos(0) = 2.

- Double Angle Identity: A trigonometric identity that expresses 2sinxcosx as sin(2x). - Chain Rule: A rule for differentiating composite functions. - Product Rule: A rule used to differentiate products of two functions. - Sine Function: A primary trigonometric function, expressed as sin(x). - Cosine Function: A primary trigonometric function, expressed as cos(x).