Find derivatives of trigonometric functions

Learning Objectives State the derivatives of the six basic trigonometric functions from memory. Apply the Product Rule to find the derivative of functions involving trigonometric components. Apply the Quotient Rule to find the derivative of functions involving trigonometric components. Apply the Chain Rule to find the derivative of composite trigonometric functions. Calculate higher-order derivatives of trigonometric functions. Use trigonometric derivatives to find the slope of a tangent line to a curve at a given point. Ever wondered how to find the exact speed of a swinging pendulum or the rate of change in an alternating current at any given instant? 🌊 The answer lies in the derivatives of the functions that model them! This tutorial will guide you through the fundament...

TermDefinitionExample DerivativeThe derivative of a function measures the instantaneous rate of change of the function with respect to one of its variables. Geometrically, it represents the slope of the tangent line to the function's graph at a specific point.If f(x) = x³, the derivative is f'(x) = 3x². Trigonometric FunctionA function (sine, cosine, tangent, etc.) that relates an angle of a right-angled triangle to ratios of its side lengths. In calculus, we use radians for the angle measure.y = sin(x), where x is an angle in radians. Chain RuleA formula to compute the derivative of a composite function. If h(x) = f(g(x)), then its derivative is h'(x) = f'(g(x)) ⋅ g'(x).The derivative of (x² + 1)³ is 3(x² + 1)² ⋅ (2x). Product RuleA formula used to find the deriv...

Derivatives of Sine and Cosine \frac{d}{dx}(\sin(x)) = \cos(x) \quad \text{and} \quad \frac{d}{dx}(\cos(x)) = -\sin(x) These are the two fundamental trigonometric derivatives from which others can be derived. Note the negative sign for the derivative of cosine. Derivatives of Tangent and Secant \frac{d}{dx}(\tan(x)) = \sec^2(x) \quad \text{and} \quad \frac{d}{dx}(\sec(x)) = \sec(x)\tan(x) These derivatives are often used together. The derivative of tangent can be found by applying the Quotient Rule to sin(x)/cos(x). Derivatives of Cotangent and Cosecant \frac{d}{dx}(\cot(x)) = -\csc^2(x) \quad \text{and} \quad \frac{d}{dx}(\csc(x)) = -\csc(x)\cot(x) Notice a pattern: the derivatives of the 'co-' functions (cosine, cotangent, cosecant) are all negative.