the derivative eric hoffman calculus plhs oct. 2007
TRANSCRIPT
The Derivative
Eric Hoffman
Calculus
PLHS
Oct. 2007
Key Topics
• Derivative: the rate of change of a function f(x) at any number x for which it is defined– Recall: rate of change really means slope– In section 2.2 we learned how to calculate the slope of
the tangent line at a point (a,b)
• The derivative of a function is another function which is denoted by f ` (f prime)
• The derivative function gives you the slope of the function f at any point along f(x)
Key Topics
By using our knowledge of calculating the slope of the line tangent to f(x) we can generalize this for all x = x0 and we can define the derivative as:
h
xfhxfh
)()(lim
0
Key Topics
Ex. Calculate the derivative of the function f(x) = x2
h
xfhxfh
)()(lim
0
h
xhxh
22
0
)(lim
h
xhhxxh
222
0
2lim
h
hhxh
2
0
2lim
)2(lim0
hxh
hh
)2(lim0
hxh
x2
xf 2
Key Topics
Calculate the derivative of the following functions:
Remember:
xxf
1)( 223)( 2 xxxf
h
xfhxfxf
h
)()(lim)(
0
xxf )(
Key Topics
Other notations for derivatives:
Note: All of the above expressions denote the derivative of y = f(x) and are read as “the derivative of ___ with respect to x” or “the derivative with respect to x”
dx
dy
x
y
)(xfx
x
f
)(xf
Rules for Calculating Derivatives
• Rule 1: if f is a constant function f(x) = c then
f `(x) = 0
• Rule 2: if f(x) = mx + b then f `(x) = m
• Rule 3 (power rule): let n ε R with n ≠ 0.
If f(x) = xn , then f `(x) = nxn-1
Properties of Derivatives
1. If the function h = f + g is differentiable, then
2. If the function r = cf, where c is a constant, is differentiable, then:
)()()( xgxfxh
)()( xfcxr
This means f `(x) exists for all x
Key Topics
Homework pg.122 1- 36 all
I will pick a few problems from section 2.3 and 2.4 that will be due on next Monday.
Word of advice, ask questions over any problem that you don’t understand, any problem that is assigned is fair game for being part of the assigment
Property 1
Find the following:
)4( 23
xxxx
)( 323 xx
x
Property 2
Find the following:
)423( 23
xxxx
)425( 323
xxx
Rule 1
Find the following:
)4(x
)9(x )3(f
Rule 2
Find the follwing:
)64(
xx
)36(
xx
)52(
xx
Rule 3
Find the following:
)( 4xx
)( 2xx
)(xx
)( xx