Derivative of cos(x) – Using Basic Differentiation Techniques and the Chain Rule

Derivative of cosx

The derivative of cos(x) can be found using basic differentiation techniques

The derivative of cos(x) can be found using basic differentiation techniques. Recall that the derivative of a function gives us the gradient (or slope) of the function at any given point.

To find the derivative of cos(x), we use the chain rule. The chain rule states that if we have a composition of functions, we need to differentiate each function separately and then multiply the derivatives together.

The derivative of cos(x) can be found as follows:

1. Start with the basic derivative of sin(x), which we know is cos(x). This is a basic trigonometric identity that can be proved.

2. Since cos(x) is the reciprocal function of sin(x), we can express it as cos(x) = 1/sin(x).

3. Next, we differentiate both sides of the equation using the quotient rule. The quotient rule states that the derivative of a quotient of functions (f(x)/g(x)) is given by (g(x)f'(x) – f(x)g'(x))/(g(x))^2.

Applying the quotient rule, we get:

d(cos(x))/dx = [d(1)/dx * sin(x) – 1 * d(sin(x))/dx] / (sin(x))^2

To simplify this further, we recall that d(1)/dx = 0 and d(sin(x))/dx = cos(x). Therefore, the derivative becomes:

d(cos(x))/dx = [-sin(x) * cos(x)] / (sin(x))^2

4. We can simplify this further by canceling out sin(x) from the numerator and denominator, leading to:

d(cos(x))/dx = -cos(x) / sin(x)

Alternatively, we can express this derivative in terms of tan(x) by using the identity sin^2(x) + cos^2(x) = 1. Dividing both sides by sin^2(x), we get 1 + cos^2(x)/sin^2(x) = csc^2(x).

Using this trigonometric identity, we can rewrite the derivative as:

d(cos(x))/dx = -cos(x) / sin(x) = -cot(x)

So, the derivative of cos(x) is -cot(x).

In summary:
d(cos(x))/dx = -cot(x)

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