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The derivative as the slope of the tangent line

(at a point)

What is a derivative?

- A function
- the rate of change of a function
- the slope of the line tangent to the curve

The tangent line

single point of intersection

slope of a secant line

f(a) - f(x)

a - x

f(x)

f(a)

x

a

slope of a (closer) secant line

f(a) - f(x)

a - x

f(x)

f(a)

a

x

x

closer and closer

a

watch the slope...

watch what x does...

x

a

The slope of the secant line gets closer and

closer to the slope of the tangent line...

As the values of x get closer and closer to a!

x

a

The slope of the secant lines gets closer to the

slope of the tangent line...

...as the values of x get closer to a

Translates to.

f(x) - f(a)

lim

x - a

a

x

as x goes to a

Equation for the slope

Which gives us the the exact slope of the line

tangent to the curve at a!

similarly...

f(xh) - f(x)

(xh) - x

f(xh) - f(x)

h

f(ah)

h

f(a)

ah

a

(For this particular curve, h is a negative value)

thus...

lim f(ah) - f(a)

h 0

h

AND

lim f(x) - f(a)

x a

x - a

Give us a way to calculate the slope of the line

tangent at a!

Which one should I use?

(doesnt really matter)

A VERY simple example...

want the slope where a2

as x a2

As h 0

back to our example...

When a2, the slope is 4

in conclusion...

- The derivative is the the slope of the line

tangent to the curve (evaluated at a point) - it is a limit (2 ways to define it)
- once you learn the rules of derivatives, you WILL

forget these limit definitions - cool site to go to for additional

explanationshttp//archives.math.utk.edu/visual.c

alculus/2/

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