Derivative of Cos y

The derivative of cos y is commonly represented as d/dy (cos y) or (cos y)', and its value is -sin(y). This indicates that the cosine function is differentiable within its domain.

Key concepts include:

Cosine Function: cos(y) is one of the fundamental trigonometric functions.

Derivative Rule: Rule for differentiating cos(y) which results in -sin(y).

Sine Function: sin(y) is another primary trigonometric function, related to cos(y).

The derivative of cos y can be denoted as d/dy (cos y) or (cos y)'.

The formula we use to differentiate cos y is: d/dy (cos y) = -sin(y) (cos y)' = -sin(y)

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

We can derive the derivative of cos y using proofs. To show this, we will use trigonometric identities and the rules of differentiation.

Here are several methods we use to prove this:

• By First Principle
   
• Using Chain Rule
   
• Using Product Rule

We will now demonstrate that the differentiation of cos y results in -sin(y) using these methods:

By First Principle

The derivative of cos y can be proved using the First Principle, expressing the derivative as the limit of the difference quotient. Consider f(y) = cos y. Its derivative can be expressed as: f'(y) = limₕ→₀ [f(y + h) - f(y)] / h Given f(y) = cos y, we write f(y + h) = cos(y + h). Substituting these into the equation: f'(y) = limₕ→₀ [cos(y + h) - cos y] / h = limₕ→₀ [-sin(y + h/2) sin(h/2)] / [h] = -limₕ→₀ [sin(h/2) / (h/2) sin(y + h/2)] Using limit formulas, limₕ→₀ [sin(h/2) / (h/2)] = 1. f'(y) = -sin(y) Hence, proved.

Using Chain Rule

To prove the differentiation of cos y using the chain rule, we use the identity: cos y = cos(y) By the chain rule: d/dy (cos y) = -sin(y)

Using Product Rule

We will now prove the derivative of cos y using the product rule. Let u = 1 and v = cos y, then d/dy (cos y) = u'v + uv'. u' = 0 and v' = -sin(y). Substituting these into the product rule gives: d/dy (cos y) = 0·cos y + 1·(-sin(y)) = -sin(y). Thus, the derivative of cos y is -sin(y).

When a function is differentiated several times, the derivatives obtained are referred to as higher-order derivatives. Higher-order derivatives provide deeper insights into the behavior of functions like cos(y). For the first derivative, we write f′(y), indicating how the function changes or its slope at a certain point.

The second derivative, f′′(y), is derived from the first derivative. Similarly, the third derivative, f′′′(y), is the result of the second derivative, and this pattern continues.

For the nth Derivative of cos(y), we use fⁿ(y) for the nth derivative, indicating the change in the rate of change.

When y is π/2, the derivative is -sin(π/2), which is -1.

When y is 0, the derivative of cos y is -sin(0), which is 0.

• Derivative: Indicates how a function changes in response to a change in the independent variable y.

• Cosine Function: A primary trigonometric function, written as cos(y).

• Sine Function: Another fundamental trigonometric function, related to cos(y).

• Chain Rule: A rule for differentiating composite functions.

• Quotient Rule: A technique for differentiating functions expressed as a ratio of two functions.