Derivative of sin(x+y)

The derivative of sin(x+y) is commonly represented as d/dx (sin(x+y)) or (sin(x+y))'. Its value is cos(x+y), indicating it is differentiable within its domain.

The key concepts are mentioned below: 

Sine Function: sin(x+y) represents the sine of the sum of two angles. 

Chain Rule: A crucial rule for differentiating composite functions like sin(x+y). 

Cosine Function: cos(x+y) is part of the derivative of sin(x+y).

The derivative of sin(x+y) can be denoted as d/dx (sin(x+y)) or (sin(x+y))'.

The formula we use to differentiate sin(x+y) is: d/dx (sin(x+y)) = cos(x+y) * (dx/dx + dy/dx)

The formula applies to all x and y where the function is defined.

We can derive the derivative of sin(x+y) using various 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

The derivative of sin(x+y) can be proved using the First Principle, which expresses the derivative as the limit of the difference quotient. To find the derivative of sin(x+y) using the first principle, consider f(x) = sin(x+y). Its derivative can be expressed as the following limit: f'(x) = limₕ→₀ [f(x + h) - f(x)] / h Given that f(x) = sin(x+y), we write f(x + h) = sin(x+h+y). Substituting these into the equation, f'(x) = limₕ→₀ [sin(x+h+y) - sin(x+y)] / h Using the identity sin(A) - sin(B) = 2 * cos((A+B)/2) * sin((A-B)/2), f'(x) = limₕ→₀ [2 * cos((2x+h+2y)/2) * sin(h/2)] / h = limₕ→₀ [cos(x+h/2+y) * sin(h/2)] / (h/2) Using limit formulas, limₕ→₀ (sin(h/2))/(h/2) = 1. f'(x) = cos(x+y) Hence, proved.

Using Chain Rule

To prove the differentiation of sin(x+y) using the chain rule, Let u = x+y, then sin(u) = sin(x+y). The derivative using the chain rule is: d/du (sin u) * du/dx = cos(u) * (d(x+y)/dx) = cos(x+y) * (1 + dy/dx) Using Product Rule We can also use the product rule to prove the derivative of sin(x+y) by considering a function of two variables. The step-by-step process is: Let u = x and v = y, then sin(x+y) = sin(u+v). Using the product rule: d/dx [sin(u+v)] = d/du [sin(u+v)] * du/dx + d/dv [sin(u+v)] * dv/dx = cos(u+v) * (1 + dy/dx) = cos(x+y) * (1 + dy/dx) Thus, the derivative is derived.

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(x+y).

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(x+y), 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+y = π/2, the derivative is zero because cos(π/2) = 0.

When x+y = 0, the derivative of sin(x+y) = cos(0), which is 1.

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

• Sine Function: A primary trigonometric function, often denoted as sin(x).

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

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

• Quotient Rule: A method used to differentiate functions that are ratios of two differentiable functions.