The derivatives of trigonometric functions are as follows:

- The derivative of the sine function (sin(x)) is the cosine function (cos(x)): d/dx(sin(x)) = cos(x).
- The derivative of the cosine function (cos(x)) is the negative sine function (-sin(x)): d/dx(cos(x)) = -sin(x).
- The derivative of the tangent function (tan(x)) is the secant squared function (sec^2(x)): d/dx(tan(x)) = sec^2(x).
- The derivative of the cosecant function (csc(x)) is the negative cosecant function multiplied by the cotangent function (-csc(x) * cot(x)): d/dx(csc(x)) = -csc(x) * cot(x).
- The derivative of the secant function (sec(x)) is the secant function multiplied by the tangent function (sec(x) * tan(x)): d/dx(sec(x)) = sec(x) * tan(x).
- The derivative of the cotangent function (cot(x)) is the negative cosecant squared function (-csc^2(x)): d/dx(cot(x)) = -csc^2(x).

These formulas represent the derivatives of the basic trigonometric functions with respect to x. They are widely used in calculus and mathematical analysis to find rates of change, slopes, and other related quantities in trigonometric equations and problems.

Here are examples demonstrating the application of the derivatives of trigonometric functions:

## 1. The derivative of the sine function (sin(x))

Example 1: Let’s find the derivative of y = sin(x).

We know that the derivative of the sine function is the cosine function. Applying this, we have: dy/dx = d/dx(sin(x)) = cos(x)

So, the derivative of y = sin(x) is dy/dx = cos(x).

## 2. The derivative of the cosine function (cos(x))

Example 2: Consider the function y = cos(x).

We know that the derivative of the cosine function is the negative sine function. Applying this, we have: dy/dx = d/dx(cos(x)) = -sin(x)

Therefore, the derivative of y = cos(x) is dy/dx = -sin(x).

## 3. The derivative of the tangent function (tan(x))

Example 3: Let’s find the derivative of y = tan(x).

We know that the derivative of the tangent function is the secant squared function. Applying this, we have: dy/dx = d/dx(tan(x)) = sec^2(x)

Thus, the derivative of y = tan(x) is dy/dx = sec^2(x).

## 4. The derivative of the cosecant function (csc(x))

Example 4: Consider the function y = csc(x).

We know that the derivative of the cosecant function is -csc(x) * cot(x). Applying this, we have: dy/dx = d/dx(csc(x)) = -csc(x) * cot(x)

Therefore, the derivative of y = csc(x) is dy/dx = -csc(x) * cot(x).

## 5. The derivative of the secant function (sec(x))

Example 5: Let’s find the derivative of y = sec(x).

We know that the derivative of the secant function is sec(x) * tan(x). Applying this, we have: dy/dx = d/dx(sec(x)) = sec(x) * tan(x)

Hence, the derivative of y = sec(x) is dy/dx = sec(x) * tan(x).

## 6. The derivative of the cotangent function (cot(x))

Example 6: Consider the function y = cot(x).

We know that the derivative of the cotangent function is -csc^2(x). Applying this, we have: dy/dx = d/dx(cot(x)) = -csc^2(x)

Thus, the derivative of y = cot(x) is dy/dx = -csc^2(x).

These examples demonstrate how to apply the derivatives of trigonometric functions to find the rate of change or slope of functions involving trigonometric equations.