Derivatives of Trigonometric Functions

The derivative of f(x) = sin(x) is f'(x) = cos(x).

Formula: d/dx (sin x) = cos x

The derivative of f(x) = cos(x) is f'(x) = -sin(x).

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

The derivative of f(x) = tan(x) is f'(x) = sec2(x).

Formula: d/dx (tan x) = sec2 x

The derivatives are:

• d/dx (csc x) = -csc x cot x
• d/dx (sec x) = sec x tan x
• d/dx (cot x) = -csc2 x

The basic derivatives (like sin x and cos x) are typically derived using the limit definition of the derivative (limh→0 [f(x + h) - f(x)] ÷ h) along with trigonometric sum formulas and special limits like limh→0 (sin h ÷ h) = 1 and limh→0 (cos h - 1) ÷ h = 0.

The derivatives of tan(x), cot(x), sec(x), and csc(x) can then be found using the Quotient Rule or Reciprocal Rule combined with the derivatives of sin(x) and cos(x) (e.g., tan x = sin x ÷ cos x).

For composite trigonometric functions, you must apply the Chain Rule. The general process is:

1. Take the derivative of the *outer* trigonometric function (treating the inside expression as a single variable).
2. Multiply the result by the derivative of the *inner* expression.

**Example:** To find the derivative of h(x) = sin(x2):

• Outer function is sin(u), where u = x². Derivative of sin(u) is cos(u).
• Inner function is x². Derivative of x² is 2x.
• Apply Chain Rule: h'(x) = cos(x2) * 2x = 2x cos(x2).