DT.01.1 - Derivative Rules -
Power Rule, Constant Rule, Sum and Difference Rule
MCB4U - Santowski

(A) Review
The equation used to find a tangent line or an instantaneous rate of change is: f ( a
 h ) lim h

0 h
 f ( a ) which we also then called a derivative.
So derivatives are calculated as .
f

( a )
 h lim

0 f ( a
 h )
 f ( a ) 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) = x 2 or 3x 2 or ax 2
 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) = x 3 or ¼ x 3 or ax 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 quadratic function f `(x) = 3ax 2

(G) Quartic Functions
(v) f(x) = x 4 or ¼ x 4 or ax 4 ?
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) = 4ax 3

(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. x 4  4x 3
GENERAL PREDICTION  x n  nx n-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) = x 2
(iv) f(x) = x 3
(v) f(x) = x 4
(vi) f(x) = x n

(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) = x n , then the derivative function f `(x) = nx n-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) = x 4 – 3x 3 + x 2 -
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) = 4x 3 –
3x 2 + 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
MCB4U – Nelson text, page 222,
Q2efghij,3efghi,5cdehi,7,8,16,13 page 229, Q2,3,9,11,13,14
IBHLY2 – Stewart, 1989, Chap 2.2, p83, Q7-10
Stewart, 1989, Chap 2.3, p88, Q1-
3eol,6-10