Dt 01 1 derivative rules power rule constant rule sum and difference rule
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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: which we also then called a derivative. So derivatives are calculated as .

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

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Dt 01 1 derivative rules power rule constant rule sum and difference rule

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

MCB4U - Santowski


A review

(A) Review

  • The equation used to find a tangent line or an instantaneous rate of change is:

  • which we also then called a derivative.

  • So derivatives are calculated as .

  • 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

(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

(C) Constant Functions


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

(D) Linear Functions


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

(E) Quadratic Functions


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

(F) Cubic Functions


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

(G) Quartic Functions


H summary

(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

(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

(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

(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


L example 11

(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

(M) Example 2


N links

(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

(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


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