$\tan x = \frac{\sin x}{\cos x}$

$\cot x = \frac{\cos x}{\sin x} = \frac{1}{\tan x}$

$\sec x = \frac{1}{\cos x}$

$\csc x = \frac{1}{\sin x}$

The instantaneous rate of change of a function.

A formula for finding the derivative of a composite function.

A function that is composed of another function. For example, $f(g(x))$.

A formula used to differentiate products of two or more functions.

A method of finding the derivative of a function that is the ratio of two other functions.

Functions that relate the angles of a triangle to the lengths of its sides.

The derivatives of co-functions (cot, csc) are negative because their original functions are decreasing over their primary intervals.

The chain rule is crucial for finding derivatives of composite functions, where one function is nested inside another. It ensures that we account for the rate of change of both the outer and inner functions.

Simplifying trig expressions before differentiating can make the process easier by reducing complexity. Use trig identities to simplify before applying derivative rules.

The unit circle provides a visual representation of trigonometric functions, helping to understand their behavior and derivatives. It shows how sine, cosine, and their related functions change as the angle varies.

The derivative of $\tan x$ is $\sec^2 x$. This means that the rate of change of the tangent function is related to the square of the secant function.

The derivative of $\cot x$ is $-\csc^2 x$. This means that the rate of change of the cotangent function is related to the negative of the square of the cosecant function.

Using radians ensures that the standard derivative formulas for trigonometric functions are valid. If degrees are used, a conversion factor is needed, making the calculations more complex.

The sign of the derivative indicates whether a function is increasing (positive derivative) or decreasing (negative derivative).

Derivatives are used to find critical points (where the derivative is zero or undefined) of a function, which can then be used to determine the maximum or minimum values of the function.

The derivative of a function gives the slope of the tangent line to the function at any given point. It represents the instantaneous rate of change of the function.

$(\tan x)' = \sec^2 x$

$(\cot x)' = -\csc^2 x$

$(\sec x)' = \sec x \tan x$

$(\csc x)' = -\csc x \cot x$

$\frac{d}{dx} f(g(x)) = f'(g(x)) cdot g'(x)$

$\frac{d}{dx} [u(x)v(x)] = u'(x)v(x) + u(x)v'(x)$

$\frac{d}{dx} \left[ \frac{u(x)}{v(x)} \right] = \frac{u'(x)v(x) - u(x)v'(x)}{[v(x)]^2}$

$\frac{d}{dx} x^n = nx^{n-1}$

$\frac{d}{dx} c = 0$

$\frac{d}{dx} cf(x) = c \frac{d}{dx} f(x)$