Derivative of cos(x/2)

We now explore the derivative of cos(x/2). It is commonly represented as d/dx (cos(x/2)) or (cos(x/2))', and its value is -1/2 sin(x/2). The function cos(x/2) is differentiable within its domain, demonstrating continuity. The key concepts are mentioned below: Cosine Function: (cos(x) = adjacent/hypotenuse). Chain Rule: Rule for differentiating composite functions like cos(x/2). Sine Function: sin(x) = opposite/hypotenuse.

The derivative of cos(x/2) can be denoted as d/dx (cos(x/2)) or (cos(x/2))'. The formula we use to differentiate cos(x/2) is: d/dx (cos(x/2)) = -1/2 sin(x/2) The formula applies to all x where the function is defined.

We can derive the derivative of cos(x/2) using different techniques. To show this, we will use trigonometric identities and differentiation rules. Here are some methods to prove this: By First Principle Using Chain Rule Using Product Rule We will demonstrate that the differentiation of cos(x/2) results in -1/2 sin(x/2) using these methods: By First Principle The derivative of cos(x/2) can be derived using the First Principle, which expresses the derivative as the limit of the difference quotient. Let's consider f(x) = cos(x/2). Its derivative can be expressed as the following limit: f'(x) = limₕ→₀ [f(x + h) - f(x)] / h … (1) Given that f(x) = cos(x/2), we write f(x + h) = cos((x + h)/2). Substituting these into equation (1), f'(x) = limₕ→₀ [cos((x + h)/2) - cos(x/2)] / h Using the cosine subtraction formula, cos(A) - cos(B) = -2 sin((A+B)/2) sin((A-B)/2), f'(x) = limₕ→₀ [-2 sin(((x + h)/2 + x/2)/2) sin(((x + h)/2 - x/2)/2)] / h = limₕ→₀ [-2 sin((2x + h)/4) sin(h/4)] / h = limₕ→₀ [-sin((2x + h)/4) sin(h/4)] / (h/2) As h approaches 0, sin(h/4)/(h/4) approaches 1, f'(x) = -sin(x/2) / 2 Therefore, f'(x) = -1/2 sin(x/2), hence proved. Using Chain Rule To prove the differentiation of cos(x/2) using the chain rule, Let u = x/2, so cos(x/2) = cos(u). The derivative of cos(u) with respect to u is -sin(u). Thus, d/dx (cos(x/2)) = (-sin(x/2)) · d/dx (x/2) = -sin(x/2) · (1/2) = -1/2 sin(x/2). Using Product Rule We can also use the product rule, but it's more straightforward with the chain rule for this function. However, rewriting cos(x/2) as a composition: cos(x/2) = cos(u) where u = x/2, Apply the chain rule as shown above.

When a function is differentiated multiple times, the resulting derivatives are higher-order derivatives. These can be complex, but are essential for understanding functions like cos(x/2). The first derivative of a function is denoted as f′(x), which indicates the slope at a particular point. The second derivative, f′′(x), is derived from the first derivative and indicates the rate of curvature. This pattern continues for third derivatives and beyond. For the nth Derivative of cos(x/2), we denote it as fⁿ(x) for the nth order, describing changes in the rate of change.

When x is such that x/2 = π/2, the derivative is undefined due to the sine function’s vertical asymptote. When x = 0, the derivative of cos(x/2) = -1/2 sin(0), which is 0.

Derivative: The derivative indicates how a function changes as its input changes. Cosine Function: A trigonometric function representing the adjacent side over the hypotenuse in a right triangle. Chain Rule: A rule used to differentiate composite functions. Sine Function: A trigonometric function representing the opposite side over the hypotenuse in a right triangle. Composite Function: A function formed by combining two functions such that the output of one function becomes the input of another.