# Derivatives and Rates of Change

## Differentiation is a way to calculate the rate of change of one variable with respect to another.

#### 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.

#### Terms

• also called gradient; slope or gradient of a line describes its steepness

#### Figures

1. ##### Slope of a function

A function with the slope shown for a given point.

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

$m = \frac{change \: in \: y}{change\: in \: x} = \frac{\triangle y}{\triangle x}$,

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 (Figure 1).

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 Glossary

curve
a simple figure containing no straight portions and no angles
##### Appears in these related concepts:
derivative
a measure of how a function changes as its input changes
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differentiation
the process of determining the derived function of a function
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e
the base of the natural logarithm, 2.718281828459045…
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function
a relation in which each element of the domain is associated with exactly one element of the co-domain
##### Appears in these related concepts:
graph
A diagram displaying data; in particular one showing the relationship between two or more quantities, measurements or indicative numbers that may or may not have a specific mathematical formula relating them to each other.
##### Appears in these related concepts:
linear
having the form of a line; straight
##### Appears in these related concepts:
mean
The average value.
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slope
also called gradient; slope or gradient of a line describes its steepness
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tangent
a straight line touching a curve at a single point without crossing it there
##### Appears in these related concepts:
variable
a quantity that may assume any one of a set of values