Derivative of All Trig Functions

The derivatives of trigonometric functions are fundamental in calculus. They help in understanding how these functions change with respect to x. The primary trigonometric functions and their derivatives are: 

Sine Function: d/dx (sin x) = cos x 

Cosine Function: d/dx (cos x) = -sin x 

Tangent Function: d/dx (tan x) = sec²x 

Cotangent Function: d/dx (cot x) = -csc²x 

Secant Function: d/dx (sec x) = sec x tan x 

Cosecant Function: d/dx (csc x) = -csc x cot x

The derivatives of the basic trigonometric functions are given by the following formulas: -

d/dx (sin x) = cos x 

d/dx (cos x) = -sin x 

d/dx (tan x) = sec²x 

d/dx (cot x) = -csc²x 

d/dx (sec x) = sec x tan x 

d/dx (csc x) = -csc x cot x

These formulas are applicable for all x within the domain of the respective trigonometric functions.

The derivatives of trigonometric functions can be derived using various methods, including trigonometric identities and rules of differentiation.

Common methods include: 

By First Principle: Using the limit definition of the derivative. 

Using Chain Rule: Applicable when functions are compositions. 

Using Quotient and Product Rules: Useful for functions that are ratios or products.

For instance, the derivative of sin x using the first principle is:

1. Consider f(x) = sin x.

2. Its derivative is f'(x) = limₕ→₀ [sin(x + h) - sin x] / h.

3. Using the identity sin(A + B) = sin A cos B + cos A sin B, we get: f'(x) = limₕ→₀ [sin x cos h + cos x sin h - sin x] / h.

4. Rearranging gives: limₕ→₀ [sin x(cos h - 1) + cos x sin h] / h.

5. Using the limits limₕ→₀ (cos h - 1)/h = 0 and limₕ→₀ sin h/h = 1, we find: f'(x) = cos x.

Similarly, other trigonometric functions can be derived using appropriate methods.

Higher-order derivatives are obtained by differentiating a function multiple times. For trigonometric functions, these derivatives often exhibit a repetitive pattern.

For example: -

• The first derivative of sin x is cos x. 
• The second derivative of sin x, which is the derivative of cos x, is -sin x. 
• The third derivative, the derivative of -sin x, is -cos x. 
• The fourth derivative, the derivative of -cos x, is sin x, completing the cycle.

Understanding these patterns helps simplify the computation of higher-order derivatives in various applications.

Certain values of x lead to special considerations in the derivatives of trigonometric functions: - At x = π/2, the derivative of cos x is 0 because cos x itself becomes 0. 

At x = 0, the derivative of sin x is 1, as sin 0 = 0 but cos 0 = 1

Some trigonometric derivatives may be undefined at specific points due to vertical asymptotes, such as the derivative of tan x at x = π/2.

• Derivative: Represents the rate of change of a function with respect to a variable.

• Trigonometric Functions: Functions related to angles, including sine, cosine, tangent, etc.

• Chain Rule: A rule for differentiating compositions of functions.

• Product Rule: A rule for differentiating products of two functions.

• Quotient Rule: A rule for differentiating quotients of two functions.