The instantaneous rate of change of a function at a given point.

The derivative of the function $f(x)$, representing the slope of the tangent line at $x$.

The rate of change of a function at a specific instant in time, equivalent to the derivative at that point.

A line that touches a curve at a single point and has the same slope as the curve at that point.

A formal way of defining the derivative as the limit of the difference quotient as h approaches zero.

A notation representing the first derivative of $y$ with respect to the independent variable (usually $x$).

Leibniz's notation for the derivative of $y$ with respect to $x$, representing an infinitesimally small change in $y$ divided by an infinitesimally small change in $x$.

How one quantity changes in relation to another quantity, often represented by the derivative.

The expression $\frac{f(x + h) - f(x)}{h}$, used to calculate the average rate of change of a function over an interval of length $h$.

The slope of the tangent line to the curve at a specific point, representing the instantaneous rate of change at that point.

$f(x)$ is increasing.

$f(x)$ is decreasing.

$f(x)$ has a horizontal tangent, which could be a local max, local min, or a stationary point.

$f(x)$ is concave up.

$f(x)$ is concave down.

$f'(x) = 0$ (a horizontal line at y=0).

$f'(x)$ does not exist at that point (non-differentiable).

The graph of $f'(x)$ is a horizontal line, representing the constant slope of $f(x)$.

$f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}$

$y - y_1 = m(x - x_1)$

$m = f'(x_0)$, where $x_0$ is the x-coordinate of the point of tangency.

1. Find $f'(x)$. 2. Evaluate $f'(x_0)$ to find the slope $m$. 3. Use point-slope form: $y - f(x_0) = m(x - x_0)$.