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
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: Separable Equations, Overview of Derivatives, and Financial Leverage
 differentiation
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 function
 Appears in these related concepts: Visualizing Domain and Range, The Vertical Line Test, and Solving Differential Equations
 graph
 Appears in these related concepts: Graphical Representations of Functions, Graphing Equations, and Graphs of Equations as Graphs of Solutions
 linear
 Appears in these related concepts: Exponential Growth and Decay, Graphs of Linear Inequalities, and Factoring General Quadratics
 mean
 Appears in these related concepts: Mean, Variance, and Standard Deviation of the Binomial Distribution, Averages, and Understanding Statistics
 real number
 Appears in these related concepts: Solving Problems with Inequalities, Introduction to Complex Numbers, and Zeroes of Polynomial Functions with Real Coefficients
 tangent
 Appears in these related concepts: Special Angles, Graphs of Exponential Functions, Base e, and Circular Motion
 variable
 Appears in these related concepts: What is a Linear Function?, Math Review, and Introduction to Variables
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