📐 Trigonometric Derivative Calculator

Mastering Trigonometric Derivatives

Trigonometric functions are essential in calculus, especially when modeling periodic phenomena like waves, oscillations, and alternating currents. Our **Trigonometric Derivative Calculator** provides a detailed, step-by-step application of the differentiation rules for all six standard trig functions: $\sin(x)$, $\cos(x)$, $\tan(x)$, $\cot(x)$, $\sec(x)$, and $\csc(x)$.

Core Trigonometric Derivative Rules

To differentiate complex expressions, the calculator intelligently combines the basic rules below with the **Chain Rule** (for compositions like $\sin(g(x))$), the **Product Rule**, and the **Quotient Rule**.

• $$\frac{d}{dx} (\sin(x)) = \cos(x)$$
• $$\frac{d}{dx} (\cos(x)) = -\sin(x)$$
• $$\frac{d}{dx} (\tan(x)) = \sec^2(x)$$
• $$\frac{d}{dx} (\cot(x)) = -\csc^2(x)$$
• $$\frac{d}{dx} (\sec(x)) = \sec(x)\tan(x)$$
• $$\frac{d}{dx} (\csc(x)) = -\csc(x)\cot(x)$$

For example, to calculate the derivative of $f(x) = \sin(x^2)$, the **Chain Rule** states $\frac{d}{dx} f(g(x)) = f'(g(x)) \cdot g'(x)$. This yields: $\cos(x^2) \cdot (2x)$. Our calculator handles this complexity instantly.

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