# The Derivative

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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)
x          x                 a
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….
lim          f(x) - f(a)
x    a          x-a

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(x+h) - f(x)
(x+h) - x

= f(x+h) - f(x)
h

f(a+h)
h

f(a)
a+h                                                    a
(For this particular curve, h is a negative value)
thus...

lim f(a+h) - 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?

(doesn’t really matter)
A VERY simple example...
yx   2

y  x2

want the slope
where a=2
f (x)  f (a)       x a
2     2
(x  a)(x  a)
lim                lim       lim
xa              xa            xa

 lim( x  a)  lim( x  2)  4

as x       a=2
f ( x  h)  f ( x )       ( x  h) 2  x 2
lim                       lim
h                        h

x  2 xh  h  x
2             2     2
h( 2 x  h)
 lim                   lim
h                   h

 lim( 2 x  h)  4

As h    0
back to our example...
yx   2

y  x2

When a=2,
the slope is 4
In conclusion...
• The derivative is the the slope of the line
tangent to the curve (evaluated at a point).
Now you know why slope was so important
• It is a limit (2 ways to define it).
• Once you learn the rules of derivatives, you
will still remember these limit definitions,
but you won’t use them to figure out limits.
Derivative of a constant function:

The derivative of f(x) = c where c is a constant is
given by  f '(x) = 0

Example   f(x) = - 10

f '(x) = 0
Derivative of a power function (power rule):
The derivative of f(x) = x r where r is a constant real
number is given by  f '(x) = r x r - 1

Example   f(x) = x -2

f '(x) = -2 x -3 = -2 / x 3
Derivative of a function multiplied by a constant:

The derivative of f(x) = c g(x) is given by  f '(x) =
c g '(x)

Example
f(x) = 3x 3 , c = 3, and g(x) = x 3

f '(x) = c g '(x) = 3 (3x 2) = 9 x 2
Derivative of the sum of functions (sum rule):
The derivative of f(x) = g(x) + h(x) is given by  f
'(x) = g '(x) + h '(x)

Example  f(x) = x 2 + 4   Let g(x) = x 2 and h(x)
= 4,
then f '(x) = g '(x) + h '(x) = 2x + 0 = 2x

```
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