Watch
Watching this resources will notify you when proposed changes or new versions are created so you can keep track of improvements that have been made.
Favorite
Favoriting this resource allows you to save it in the “My Resources” tab of your account. There, you can easily access this resource later when you’re ready to customize it or assign it to your students.
Derivatives and Rates of Change
Differentiation is a way to calculate the rate of change of one variable with respect to another.
Learning Objective

Describe the derivative as the change in
$y$ over the change in$x$ at each point on a graph
Key Points
 Historically, the primary motivation for the study of differentiation was the tangent line problem, which is the task of, for a given curve, finding the slope of the straight line that is tangent to that curve at a given point.
 If
$y$ is a linear function of$x$ , then$m = \frac{\Delta y}{\Delta x}$ .  The derivative measures the slope of a graph at each point.
Term

slope
also called gradient; slope or gradient of a line describes its steepness
Full Text
Historically, the primary motivation for the study of differentiation was the tangent line problem, which is the task of, for a given curve, finding the slope of the straight line that is tangent to that curve at a given point. The word tangent comes from the Latin word tangens, which means touching. Thus, to solve the tangent line problem, we need to find the slope of a line that is "touching" a given curve at a given point, or, in modern language, that has the same slope. But what exactly do we mean by "slope" for a curve?
The simplest case is when
where the symbol
It follows that
Slope of a function
A function with the slope shown for a given point.
This gives an exact value for the slope of a straight line. If the function
Assign just this concept or entire chapters to your class for free.
Key Term Reference
 curve
 Appears in these related concepts: Arc Length and Surface Area, Arc Length and Speed, and Area Between Curves
 derivative
 Appears in these related concepts: Integration By Parts, Separable Equations, and Overview of Derivatives
 differentiation
 Appears in these related concepts: Development of Nervous Tissue, Caulobacter Differentiation, and The Challenge of Competition
 function
 Appears in these related concepts: Inverse Functions, Solving Differential Equations, and Functions and Their Notation
 graph
 Appears in these related concepts: Graphing on Computers and Calculators, Reading Points on a Graph, and Graphical Representations of Functions
 linear
 Appears in these related concepts: SecondOrder Linear Equations, Linear Approximation, and Graphs of Linear Inequalities
 mean
 Appears in these related concepts: Mean, Variance, and Standard Deviation of the Binomial Distribution, The Mean Value Theorem, Rolle's Theorem, and Monotonicity, and Understanding Statistics
 real number
 Appears in these related concepts: Graphing the Normal Distribution, Real Numbers, Functions, and Graphs, and Solving Problems with Inequalities
 tangent
 Appears in these related concepts: Derivatives of Exponential Functions, Direction Fields and Euler's Method, and Tangent Vectors and Normal Vectors
 variable
 Appears in these related concepts: Related Rates, Controlling for a Variable, and Math Review
Sources
Boundless vets and curates highquality, openly licensed content from around the Internet. This particular resource used the following sources:
Cite This Source
Source: Boundless. “Derivatives and Rates of Change.” Boundless Calculus. Boundless, 26 May. 2016. Retrieved 28 May. 2016 from https://www.boundless.com/calculus/textbooks/boundlesscalculustextbook/derivativesandintegrals2/derivatives9/derivativesandratesofchange432928/