Rate of Change and Derivative Math 1231: Single-Variable Calculus

Tangent Line

Examples Example: Find an equation of the tangent line to y = sin(x) at the point P(0, 0).

Another Expression

Velocity Instantaneous velocity

Rate of Change Suppose y is a quantity that depends on another quantity x. If x changes from x 1 to x 2, the then change in x (also called the increment of x) is Δx = x 2 – x 1 and the corresponding change in y is Δy = f(x 2 ) – f(x 1 ) The difference quotient Δy/Δx is called the average rate of change of y with respect to x over the interval [x 1, x 2 ]. The limit of the average rates of change as x 2 approaches x 1 is called the (instantaneous) rate of change of y with respect to x at x = x 1.

Derivative Example: Find the derivative of f(x) = x 2 at the number a. Example: Find the tangent line to f(x) = x 2 at the number a.