Transcript DT.01.1 - Derivative Rules - Power Rule, Constant Rule

B1.1 & B2.1 – Derivatives of
Power Functions - Power Rule,
Constant Rule, Sum and
Difference Rule
IBHL/SL Y2 - Santowski
(A) Review
The equation used to find a tangent line or an
instantaneous rate of change is:
f ( a  h)  f ( a )
lim
h 0
h
which we also then called a derivative.
So derivatives are calculated as
f ( a  h)  f ( a )
.f (a )  lim
h 0
h
Since we can differentiate at any point on a function, we
can also differentiate at every point on a function (subject
to continuity)  therefore, the derivative can also be
understood to be a function itself.
(B) Finding Derivatives Without
Using Limits – Differentiation Rules
We will now develop a variety of useful
differentiation rules that will allow us to calculate
equations of derivative functions much more
quickly (compared to using limit calculations each
time)
First, we will work with simple power functions
We shall investigate the derivative rules by
means of the following algebraic and GC
investigation (rather than a purely “algebraic”
proof)
(C) Constant Functions
(i) f(x) = 3 is called a
constant function  graph
and see why.
What would be the rate of
change of this function at x
= 6? x = -1, x = a?
We could do a limit
calculation to find the
derivative value  but we
will graph it on the GC and
graph its derivative.
So the derivative function
equation is f `(x) = 0
(D) Linear Functions
(ii) f(x) = 4x or f(x) = -½x
or f(x) = mx  linear fcns
What would be the rate of
change of this function at x
= 6? x = -1, x = a.
We could do a limit
calculation to find the
derivative value  but we
will graph it on the GC and
graph its derivative.
So the derivative function
equation is the constant
function f `(x) = m
(E) Quadratic Functions
(iii) f(x) = x2 or 3x2 or ax2
 quadratics
What would be the rate of
change of this function at x
= 6? x = -1, x = a.
We could do a limit
calculation to find the
derivative value  but we
will graph it on the GC and
graph its derivative.
So the derivative function
equation is the linear
function f `(x) = 2ax
(F) Cubic Functions
(iv) f(x) = x3 or ¼ x3 or
ax3?
What would be the rate of
change of this function at x
= 6? x = -1, x = a.
We could do a limit
calculation to find the
derivative value  but we
will graph it on the GC and
graph its derivative.
So the derivative function
equation is the quadratic
function f `(x) = 3ax2
(G) Quartic Functions
(v) f(x) = x4 or ¼ x4 or
ax4?
What would be the rate of
change of this function at x
= 6? x = -1, x = a.
We could do a limit
calculation to find the
derivative value  but we
will graph it on the GC and
graph its derivative.
So the derivative function
equation is the cubic
function f `(x) = 4ax3
(H) Summary
We can summarize the observations
from previous slides  the derivative
function is one power less and has
the coefficient that is the same as
the power of the original function
i.e. x4  4x3
GENERAL PREDICTION  xn  nxn-1
(I) Algebraic Verification
do a limit calculation on the following
functions to find the derivative functions:
(i) f(x) = k
(ii) f(x) = mx
(iii) f(x) = x2
(iv) f(x) = x3
(v) f(x) = x4
(vi) f(x) = xn
(J) Power Rule
We can summarize our findings in a “rule”  we
will call it the power rule as it applies to power
functions
If f(x) = xn, then the derivative function f `(x) =
nxn-1
Which will hold true for all n
(Realize that in our investigation, we simply
“tested” positive integers  we did not test
negative numbers, nor did we test fractions, nor
roots)
(K) Sum, Difference, Constant
Multiple Rules
We have seen derivatives of simple
power functions, but what about
combinations of these functions?
What is true about their derivatives?
(L) Example #1
(vi) f(x) = x² + 4x - 1
What would be the rate of
change of this function at x
= 6? x = -1, x = a.
We could do a limit
calculation to find the
derivative value  but we
will graph it on the GC and
graph its derivative.
So the derivative function
equation is the “combined
derivative” of f `(x) = 2x +
4
(L) Example #1
(M) Example 2
(vii) f(x) = x4 – 3x3 + x2 ½x + 3
What would be the rate of
change of this function at x
= 6? x = -1, x = a.
We could do a limit
calculation to find the
derivative value  but we
will graph it on the GC and
graph its derivative.
So the derivative function
equation is the “combined
derivative” f `(x) = 4x3 –
3x2 + 2x – ½
(N) Links
Visual Calculus - Differentiation Formulas
Calculus I (Math 2413) - Derivatives Differentiation Formulas from Paul
Dawkins
Calc101.com Automatic Calculus featuring
a Differentiation Calculator
Some on-line questions with hints and
solutions
(O) Homework
IBHL/SL Y2 – Stewart, 1989, Chap
2.2, p83, Q7-10
Stewart, 1989, Chap 2.3, p88, Q13eol,6-10