Download Derivative of a Function

Ch. 3 – Derivatives
3.1 – Derivative of a
Function
• The rate of change, or slope, of a function is
called its derivative. It is denoted by f’(x),
which is read as “f prime of x”.
f ( x  h)  f ( x )
f '( x)  lim
h 0
h
– The derivative is an equation for the slope of the
tangent line at any point (x, f(x)).
– If f’(x) exists for some value x, then we say f is
differentiable at x.
– A function differentiable at every point in its
domain is a differentiable function.
• The derivative of f at x=a can also be found
by…
f ( x)  f (a)
f '(a)  lim
x a
xa
• Ex: Find the derivative of f(x)=4x2 when x=2.
– Method 1:
4( x  h) 2  4 x 2
f ( x  h)  f ( x )
 lim
f '( x)  lim
h 0
h 0
h
h
4 x 2  8 xh  4h 2  4 x 2
 lim
h 0
h
 8x
x2
8 xh  4h 2
 lim
h 0
h
 lim(8 x  4h)
h 0
 16
– Method 2:
2
f ( x)  f (2)
4
x
 16
f '(2)  lim
 lim
x2
x  (2)
x2
x2
4( x  2)( x  2)
 lim
x2
x2
 lim 4( x  2)
x2
4( x 2  4)
 lim
x2
x2
 16
• The following symbols indicate the
derivative of a function y=f(x). THEY ALL
MEAN THE SAME THING!
y'
f'
» Read as “y prime”
» “f prime”
dy
dx
» “dy dx” or “the derivative of y with
respect to x”
df
dx
» “df dx”
d
f ( x)
dx
» “d dx of f at x” or “the derivative of f
at x”
Graphing f’ from f
• Graph the derivative of the function f
shown below. Use key points to generate
the graph.
f(x)
++
–
+
Step 1: Identify zeros (where slope is a horizontal line)
Step 2: Identify positive/negative slope ranges between zeros
Step 3: Identify how positive/negative slope will be
Step 4: Graph the derivative
Graph the Derivative!
Graph the Derivative!
Alternate Def’n for Differentiability
(3.2)
• If f(x) is continuous at x=a, then f(x) is
differentiable at a if…
f ( x)  f (a)
f ( x)  f (a)
lim
 lim
xa
xa
xa
xa
 2 x, x  0
g ( x)  
 x , x  0
• Ex: Is g(x) differentiable over the real numbers?
– g(x) is definitely differentiable for every value besides zero,
so lets check the left and right derivatives at zero.
lim
g ( x)  g (0)
2x  0
2x
 lim
 lim
x 0 x  0
x0
x0
x
lim
g ( x)  g (0)
x 0
x
1
 lim
 lim
 lim
x 0
x0
x 0
x0
x0
x
x
x 0
x 0
 lim 2
x 0
2
 does not exist
– Since the derivatives to the left and right of zero aren’t
equal, g(x) is not differentiable at x=0.