Elementary Algebra
Introduction to Algebra Linear Equations and Inequalities Functions and Graphs I Lines and thier Graphs Linear Systems Exponents & Polynomials
Intermediate Algebra
Factoring Rational Expressions Rational Equations and Applications Radical Expressions Nonlinear Equations and Applications Functions and Graphs II Exponential and Logarithmic Functions
Precalculus I / College Algebra
Equations and Inequalitites Functions and Graphs Polynomial and Rational Functions Exponential and Logarithmic Functions Systems and Matrices Geometry Basics Conic Sections Sequences and Series
Precalculus II / Trigonometry
The Six Trigonometric Functions Right Triangle Trigonometry Circular Functions Graphs of Trigonometric Functions Trigonometric Identities Trigonometric Equations Oblique Triangles and the Laws Vectors Complex, Parametric, and Polar Forms
Calculus I
Limits and Continuity Derivatives Analysis of Curves Antiderivatives
Calculus II
Transcendental Functions
Geometry Physics Integration Techniques Calculus of Infinity Parametric, Polar, and Conic Curves
Calculus III

Course: Calculus I
Topic: Derivatives
Subtopic: Product Rule, Quotient Rules, & Deriv. of Trig Functions

Overview

If we wish to take a derivative of, for example, x/sinx, we need a rule for taking the derivative of a quotient. Not surprisingly, there is one, and it is called the quotient rule. That along with the product rule, which finds the derivative of a produce such as xex, allow up to find the derivatives of the remaining four trigonometric functions.

Objectives

By the end of this topic you should know and be prepared to be tested on:

• 2.4.1 Apply the product rule to differentiate a product
• 2.4.2 Apply the quotient rule to differentiate a quotient
• 2.4.3 Know the derivative of all six trigonometric functions
• 2.4.4 Find the slope and equation of a tangent line to a function involving a product or quotient function using the derivative without using the formal definition of derivative

Terminology

Terms you should be able to define: product rule, quotient rule

 Product Rule d/dx(f(x)*g(x))=f(x)*g'(x)+f'(x)*g(x) Or as a mnemonic: (F*L)'=F*L'+F'*L where F = first function and L = last function Quotient Rule d/dx(f(x)/g(x))=(f(x)*g'(x)-f'(x)*g(x))/g^2(x) Or as a mnemonic: (N/D)'=(D*N'-N*D')/D^2 where N = numerator, D = denominator

Derivatives of the Trigonometric Functions

 d/dx(sinx) = cosx d/dx(tanx) = sec^2x d/dx(secx) = secx tanx d/dx(cosx) = -sinx d/dx(cotx) = -csc^2x d/dx(cscx) = -cscx cotx

Text Notes

• The product rule and quotient rule are important to memorize! There are songs that serve as mnemonics on YouTube if you want to check them out. For example: Hi-Dee-Lo Song

• Be sure that you can apply the quotient rule to the ratio identities for the trig functions tanx, cotx, secx, and cscx to find their derivatives.

Mini-Lectures and Examples

Supplemental Resources (optional)

rev. 2020-10-10