The derivatives of trig functions are a core part of solving many Calculus problems! They are listed below:

\[ \begin{align*} \dfrac{d}{dx} \sin x &= \cos x\\ \dfrac{d}{dx} \cos x &= -\sin x\\ \dfrac{d}{dx} \tan x &= \sec^2 x\\ \dfrac{d}{dx} \cot x &= – \csc^2 x \\ \dfrac{d}{dx} \sec x &= \sec x \tan x\\ \dfrac{d}{dx} \csc x &= -\csc x \cot x\\ \end{align*} \]

Tips for Remembering:

• Sine and Cosine: The derivative of $ \sin x $ is $\cos x $, and the derivative of  $\cos x $ is $ -\sin x $. Think of Sine as staying the same and Cosine as changing..
• Tangent and Secant: $\tan x$ pairs with $\sec^2 x$ and they are linked by the Pythagorean identity $1+\tan^{⁡2} x=\sec^{⁡2} x$. Similarly, $\cot x $ pairs up with $-\csc x$, and they are linked by the Pythagorean identity $1+\cot^{⁡2} x=\csc^{⁡2} x$.
• Cosine, Cosecant and Cotangent: All Co-functions derivatives contain negative signs, which helps group them together mentally.

Key Applications:

• Chain Rule: These formulas are often used with the chain rule. For example, if $y=\sin(3x)$, then $\dfrac{dy}{dx}=3\cos⁡(3x)$.
• Solving Equations and Modeling: Trigonometric derivatives are essential in problems involving circular motion, geometry and oscillations, making them useful in physics, engineering, and signal processing.

By remembering the patterns in these derivatives and their relationships, you can more easily apply them to complex calculus problems.