Differentiation of trigonometric functions

Differentiation of trigonometric functions gives formulas for their rates of change, such as the derivatives of sine and cosine, and rules for differentiating their compositions.

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Derivative of sine: The derivative of the sine function is the cosine function. This is the basic rate-of-change rule for sine.
Derivative: A derivative measures a function’s instantaneous rate of change and gives the slope of its graph at a point. Trigonometric differentiation is a collection of derivative rules for specific functions.
Related rates: Related rates problems use differentiation to connect changing quantities linked by an equation. Trigonometric relations often connect changing angles to lengths in geometric motion.
Differentiation of exponential functions: Differentiation of exponential functions gives rate-of-change rules for functions whose variable appears in an exponent. Exponential derivatives have different defining behavior, though they connect to trigonometric functions through complex numbers.
Derivative of cosine: The derivative of the cosine function is the negative sine function. The negative sign distinguishes cosine’s rate of change from sine’s.
Limit (mathematics): A limit describes the value a function approaches as its input approaches a specified value. The defining limits for sine and cosine establish their derivative formulas.
Simple harmonic motion: Simple harmonic motion is periodic motion in which acceleration is proportional to displacement and directed toward equilibrium. Sine and cosine derivatives connect displacement, velocity, and acceleration.
Differentiation of hyperbolic functions: Differentiation of hyperbolic functions gives rate-of-change rules for sinh, cosh, tanh, and related functions. Hyperbolic derivatives resemble trigonometric formulas but differ in key signs and identities.
Derivative of tangent: The derivative of tangent is secant squared wherever tangent is defined. It follows from the quotient rule or from tangent’s relation to sine and cosine.
Radian: A radian is an angle measure equal to the ratio of an arc’s length to its radius. The standard derivatives of sine and cosine hold in this simple form when angles are measured in radians.