## Derivative of Cos

### The derivative of the cosine function, denoted as d(cos(x))/dx or cos'(x), is found using the rules of calculus

The derivative of the cosine function, denoted as d(cos(x))/dx or cos'(x), is found using the rules of calculus. The derivative represents the rate of change of a function with respect to its variable.

The derivative of the cosine function is -sin(x). In other words, the derivative of cos(x) with respect to x is -sin(x). This means that if you were to plot the cosine function on a graph, the derivative of the function at any point x would give you the slope of the tangent line to the graph at that point.

To explain how we arrive at this result, we can use the definition of the derivative and some trigonometric identities. The definition of the derivative of a function f(x) with respect to x is:

f'(x) = lim(h->0) [f(x+h) – f(x)] / h

Applying this definition to the cosine function, we have:

cos'(x) = lim(h->0) [cos(x+h) – cos(x)] / h

Now, we can use the trigonometric identity for the cosine of a sum of angles:

cos(a + b) = cos(a)cos(b) – sin(a)sin(b)

Applying this identity to our expression:

cos(x + h) = cos(x)cos(h) – sin(x)sin(h)

Now let’s substitute this back into our derivative expression:

cos'(x) = lim(h->0) [(cos(x)cos(h) – sin(x)sin(h)) – cos(x)] / h

Simplifying further:

cos'(x) = lim(h->0) [(cos(x)cos(h) – cos(x)) – sin(x)sin(h)] / h

cos'(x) = lim(h->0) [cos(x)(cos(h) – 1) – sin(x)sin(h)] / h

Now, as h approaches 0, we can make some simplifications by applying the limit laws:

cos'(x) = cos(x) * lim(h->0) (cos(h) – 1) / h – sin(x) * lim(h->0) sin(h) / h

The limits of (cos(h) – 1) / h and sin(h) / h as h approaches 0 can be evaluated using the concept of limits in calculus. The limit of (cos(h) – 1) / h as h approaches 0 is 0, and the limit of sin(h) / h as h approaches 0 is 1.

Therefore, the final result is:

cos'(x) = cos(x) * 0 – sin(x) * 1

cos'(x) = -sin(x)

Hence, the derivative of the cosine function is -sin(x).

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