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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: for a given curve: find the slope of the straight line that is tangent to the curve at a given point.
 If y is a linear function of x, then
$m = \frac{change \: in \: y}{change\: in \: x} = \frac{\triangle y}{\triangle x}$ .  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: for a given curve: find the slope of the straight line that is tangent to the 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 y is a linear function of x, meaning that the graph of y divided by x is a straight line. In this case, y = f(x) = m x + b, for real numbers m and b, and the slope m is given by
where the symbol Δ (the uppercase form of the Greek letter Delta) is an abbreviation for "change in. " This formula is true because y + Δy = f(x+ Δx) = m (x + Δx) + b = m x + b + m Δx = y + mΔx.It follows that Δy = m Δx .
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 f is not linear (i.e. its graph is not a straight line), however, then the change in y divided by the change in x varies: differentiation is a method to find an exact value for this rate of change at any given value of x. In other words, differentiation is a method to compute the rate at which a dependent output y changes with respect to the change in the independent input x. This rate of change is called the derivative of y with respect to x. In more precise language, the dependence of y upon x means that y is a function of x. This functional relationship is often denoted y = f(x), where f denotes the function. If x and y are real numbers, and if the graph of y is plotted against x, the derivative measures the slope of this graph at each point.
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: Exponential and Logarithmic Functions, Antiderivatives, and Defining Financial Leverage
 differentiation
 Appears in these related concepts: Considering the Environment, Defining Monopoly, and Socioemotional Development in Adolescence
 function
 Appears in these related concepts: Inverse Functions, Solving Differential Equations, and Modal Mixture
 graph
 Appears in these related concepts: Graphing on Computers and Calculators, Reading Points on a Graph, and Graphing Functions
 linear
 Appears in these related concepts: Trinomials of the Form ax^2 + bx + c, Where a is Not Equal to 1, Exponential Growth and Decay, 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, Intermediate Value Theorem, and The ComplexNumber System
 tangent
 Appears in these related concepts: Newton's Method, Graphs of Exponential Functions, Base e, and Circular Motion
 variable
 Appears in these related concepts: Related Rates, Math Review, and Psychology and the Scientific Method: From Theory to Conclusion
Sources
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Cite This Source
Source: Boundless. “Derivatives and Rates of Change.” Boundless Calculus. Boundless, 21 Jul. 2015. Retrieved 23 Jul. 2015 from https://www.boundless.com/calculus/textbooks/boundlesscalculustextbook/derivativesandintegrals2/derivatives9/derivativesandratesofchange432928/