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...
yx 2
y x2
want the slope
where a=2
f (x) f (a) x a
2 2
(x a)(x a)
lim lim lim
xa xa xa
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...
yx 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
in your algebra classes!
• 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