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Derivative as a Limit of Slopes

The derivative of the function f at x = a is the slope of the function f at x = a.What the wha—? We typically find the slope of a...

Derivatives as an Instantaneous Rate of Change

Approach the derivative from a different and more real-lifey direction.There are two ways to think about speed while driving. The first way is to look at the speedometer, which shows how fast the...

Tangent Lines

Daisy, our favorite track star, is learning how to throw discus via Youtube videos. She wants to know exactly when to release the discus so that it flies off in the right direction. ...

Tangent Line Approximation

There are only two things we need to remember about the tangent line to f at a. The tangent line and f have the same y-value at a.     That is, the point (a,f(a)) is on f and a...

Differentiability and Continuity

We say a function is differentiable at a if f ' (a) exists. A function is differentiable on an interval if f ' (a) exists for every value of a in the interval. We say a function is diffe...

The Derivative Function

The "derivative of f at a," written f ' (a), is a number that is equal to the slope of the function f at a.For any differentiable function f there is another function, known as the derivative...

Graphs of f ( x ) and f ' ( x )

From a graph of a function f(x) we can make a sketched graph of its derivative f ' (x). To do this, we use some things we talked about earlier.If f is decreasing, its slope (and hence its der...

Theorems

There are several important theorems that help to describe derivatives in calculus. A lot of the time, looking a the curve of a function that is described will help us literally see what is going o...

In the Real World

Derivatives are, believe it or not, used a lot. The field of differential equations is an area of math that studies equations with derivatives in them.Differential equations are used to investigate...
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