the introduction section, we talked about the average rate of change of
a function. What happens as you let the "h" value approach
zero? Why, of course you are talking about a limit! Look
what happens to the line as h approaches zero!
The red curve is the function
and the blue line is the slope at a given point! Does this line
cross the function twice like the previous section? No way dude! It
only crosses the function once. What is a line that crosses a
circle exactly once called? If you don't know, ask Mrs.
Parker!! That's right, it is a tangent
line!! What you have found by
letting h approach zero, is the slope of
the tangent line!! Important topic
The slope of the tangent line =
Which is denoted f ' (x) and is called the derivative of f!!
Find f'(x) if f(x) = x2.
f'(x) = 2x for any point on the curve of x2.
find the slope at the point x = 2 and then at x = -3
find the following:
(2) = 2(2) = 4
(-3) = 2(-3) = -6
that the slope at x = 2 is 4 and the slope at x = -3 is -6. Both
these make sense when you consider at x = 2 the graph of the parabola
increasing and at x = -3 it is decreasing!
To find f
'(x) by the definition becomes very tedious work. Therefore, we
have some simple formulas to find the derivatives of various
functions. Each of
these formulas will be proved in class.
1) If f(x) = c, where c is a constant, then f '(x) = 0.
This makes sense when you
consider a constant graph is a horizontal line and all horizontal lines
have a slope of zero!
2) If f(x) = xn, then f '(x) = nxn - 1.
f(x) = x2,
f '(x) = 2x
f(x) = x5, f
'(x) = 5x4.
f(x) = x-4,
f '(x) = -4x-5.
f(x) = x1/2,
f '(x) = 1/2(x)-1/2.
to do. Bring the power out front and decrease the power by one!
f(x) = cxn,
then f '(x) = cnxn - 1.
f(x) = 3x2,
f '(x) = 6x.
f(x) = -4x3,
f '(x) = -12x2.
4) If f(x) = p(x) + q(x), then f '(x) = p '(x) + q '(x).
f(x) = 3x2 +
1, f '(x) = 6x + 4 (remember, the derivative of a constant is zero!)
f(x) = x3 + 4x2 - 5x + 3, f '(x) = 3x2 + 8x - 5.
the slope at x =1
'(1) = 3(1)2 +
8(1) - 5 = 3 + 8 - 5 = 6
Problems to try
Find the first derivative
of each and find the slope at x = 2