The derivative

Elaboración: Dra. Diana Denys Jiménez Suro Dr. Antonio Jesús Sánchez Hernández Campus Estado de México Campus Santa Fe

The derivative

Mathematical Thinking I

The tangent

The word "tangent" is derived from the Latin word "tangens," which means "to touch." Thus, a tangent to a curve is a straight line that touches the curve. In other words, a tangent line must have the same direction as the curve at the point of contact, but how can we mathematically express this idea?

The derivative as a rate of change

Given any function 𝑦 = ƒ(𝑥), we calculate the average rate of change of 𝑦 with respect to 𝑥 over the interval [𝑥1, 𝑥2] by dividing the change in the value of 𝑦, ∆𝑦 = ƒ(𝑥2) − ƒ(𝑥1), by the length ∆𝑥 = 𝑥2 − 𝑥1 = ℎ of the interval over which the change occurs.

Derivative at a point

If we introduce the concept of a limit to the rate of change, that is, if we let h approach zero and the limit exists, then we obtain the derivative of a function ƒ at a point 𝑥0, denoted ƒ′(𝑥0), is

Desmos

Slope of the tangent line

Given a function 𝑓(𝑥) and a point 𝑥=𝑎, the derivative function 𝑓′(𝑥) indicates the slope of the tangent line at that point, and the value of the slope is 𝑓′(𝑎). The equation of the tangent line is:

Desmos

Let's consider the following differentiation formulas; the use of these formulas simplifies the process when calculating derivatives, eliminating the need to compute them through limits.

Basic differentiation formulas

If we have two functions 𝑓(𝑥) and 𝑔(𝑥), as well as a constant 𝑐, the following differentiation rules apply.

Differentiation rules

Exercises

Quiz

Canvas

Stewart

Calculus Early Transcendentals8a. Edición, CENGAGE

Thomas

Cálculus Multivariable13a. Edición, PEARSON

Thomas

2018

2015

2015

References

Calculus Early Transcendentals13a. Edición, PEARSON

Antonio Jesús Sánchez Hernández Campus Santa Fe ajsanchez@tec.mx

Diana Denys Jiménez Suro Campus Estado de México ddjimenez@tec.mx

Teachers