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

and Difference Rule and Difference Rule

MCB4U - SantowskiMCB4U - Santowski

(A) Review(A) ReviewThe equation used to find a tangent line or an The equation used to find a tangent line or an instantaneous rate of change is:instantaneous rate of change is:

which we also then called a derivative.which we also then called a derivative.

So derivatives are calculated as .So derivatives are calculated as .

Since we can differentiate at any point on a function, we Since we can differentiate at any point on a function, we can also differentiate at every point on a function (subject can also differentiate at every point on a function (subject to continuity) to continuity) therefore, the derivative can also be therefore, the derivative can also be understood to be a function itself.understood to be a function itself.

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(B) Finding Derivatives Without (B) Finding Derivatives Without Using Limits – Differentiation RulesUsing Limits – Differentiation Rules

We will now develop a variety of useful We will now develop a variety of useful differentiation rules that will allow us to calculate differentiation rules that will allow us to calculate equations of derivative functions much more equations of derivative functions much more quickly (compared to using limit calculations each quickly (compared to using limit calculations each time)time)

First, we will work with simple power functionsFirst, we will work with simple power functions

We shall investigate the derivative rules by We shall investigate the derivative rules by means of the following algebraic and GC means of the following algebraic and GC investigation (rather than a purely “algebraic” investigation (rather than a purely “algebraic” proof)proof)

(C) Constant Functions(C) Constant Functions(i) f(x) = 3 is called a (i) f(x) = 3 is called a constant function constant function graph graph and see why.and see why.What would be the rate of What would be the rate of change of this function at change of this function at xx = 6? = 6? xx = -1,  = -1, xx = = a?a?We could do a limit We could do a limit calculation to find the calculation to find the derivative value derivative value but we but we will graph it on the GC and will graph it on the GC and graph its derivative.graph its derivative.So the derivative function So the derivative function equation is f `(x) = 0equation is f `(x) = 0

(D) Linear Functions(D) Linear Functions(ii) f(x) = 4x or f(x) = -½x (ii) f(x) = 4x or f(x) = -½x or f(x) = mx or f(x) = mx linear fcns linear fcns What would be the rate of What would be the rate of change of this function at change of this function at xx = 6? = 6? xx = -1,  = -1, xx = = aa. . We could do a limit We could do a limit calculation to find the calculation to find the derivative value derivative value but we but we will graph it on the GC and will graph it on the GC and graph its derivative.graph its derivative.So the derivative function So the derivative function equation is the constant equation is the constant function f `(x) = mfunction f `(x) = m

(E) Quadratic Functions(E) Quadratic Functions(iii) f(x) = x(iii) f(x) = x22 or 3x or 3x22 or ax or ax22 quadratics quadratics What would be the rate of What would be the rate of change of this function at change of this function at xx = 6? = 6? xx = -1,  = -1, xx = = aa. . We could do a limit We could do a limit calculation to find the calculation to find the derivative value derivative value but we but we will graph it on the GC and will graph it on the GC and graph its derivative.graph its derivative.So the derivative function So the derivative function equation is the linear equation is the linear function f `(x) = 2axfunction f `(x) = 2ax

(F) Cubic Functions(F) Cubic Functions(iv) f(x) = x(iv) f(x) = x33 or ¼ x or ¼ x33 or ax or ax33? ? What would be the rate of What would be the rate of change of this function at change of this function at xx = 6? = 6? xx = -1,  = -1, xx = = aa. . We could do a limit We could do a limit calculation to find the calculation to find the derivative value derivative value but we but we will graph it on the GC and will graph it on the GC and graph its derivative.graph its derivative.So the derivative function So the derivative function equation is the quadratic equation is the quadratic function f `(x) = 3axfunction f `(x) = 3ax22

(G) Quartic Functions(G) Quartic Functions(v) f(x) = x(v) f(x) = x44 or ¼ x or ¼ x44 or ax or ax44? ? What would be the rate of What would be the rate of change of this function at change of this function at xx = 6? = 6? xx = -1,  = -1, xx = = aa. . We could do a limit We could do a limit calculation to find the calculation to find the derivative value derivative value but we but we will graph it on the GC and will graph it on the GC and graph its derivative.graph its derivative.So the derivative function So the derivative function equation is the cubic equation is the cubic function f `(x) = 4axfunction f `(x) = 4ax33

(H) Summary(H) SummaryWe can summarize the observations We can summarize the observations from previous slides from previous slides the derivative the derivative function is one power less and has function is one power less and has the coefficient that is the same as the coefficient that is the same as the power of the original functionthe power of the original function

i.e. xi.e. x44 4x 4x33

GENERAL PREDICTION GENERAL PREDICTION x xnn nx nxn-1n-1

(I) Algebraic Verification(I) Algebraic Verificationdo a limit calculation on the following do a limit calculation on the following functions to find the derivative functions:functions to find the derivative functions:

(i) f(x) = k(i) f(x) = k

(ii) f(x) = mx(ii) f(x) = mx

(iii) f(x) = x(iii) f(x) = x22

(iv) f(x) = x(iv) f(x) = x33

(v) f(x) = x(v) f(x) = x44

(vi) f(x) = x(vi) f(x) = xnn

(J) Power Rule(J) Power RuleWe can summarize our findings in a “rule” We can summarize our findings in a “rule” we we will call it the power rule as it applies to power will call it the power rule as it applies to power functionsfunctions

If f(x) = xIf f(x) = xnn, then the derivative function f `(x) = , then the derivative function f `(x) = nxnxn-1n-1

Which will hold true for all nWhich will hold true for all n(Realize that in our investigation, we simply (Realize that in our investigation, we simply “tested” positive integers “tested” positive integers we did not test we did not test negative numbers, nor did we test fractions, nor negative numbers, nor did we test fractions, nor roots)roots)

(K) Sum, Difference, Constant (K) Sum, Difference, Constant Multiple RulesMultiple Rules

We have seen derivatives of simple We have seen derivatives of simple power functions, but what about power functions, but what about combinations of these functions? combinations of these functions? What is true about their derivatives?What is true about their derivatives?

(L) Example #1(L) Example #1(vi) f(x) = x² + 4x - 1 (vi) f(x) = x² + 4x - 1

What would be the rate of What would be the rate of change of this function at change of this function at xx = 6? = 6? xx = -1,  = -1, xx = = aa. . We could do a limit We could do a limit calculation to find the calculation to find the derivative value derivative value but we but we will graph it on the GC and will graph it on the GC and graph its derivative.graph its derivative.So the derivative function So the derivative function equation is the “combined equation is the “combined derivative” of f `(x) = 2x + derivative” of f `(x) = 2x + 44

(L) Example #1(L) Example #1

(M) Example 2(M) Example 2(vii) f(x) = x(vii) f(x) = x44 – 3x – 3x33 + x + x22 -½x -½x + 3 + 3 What would be the rate of What would be the rate of change of this function at change of this function at xx = 6? = 6? xx = -1,  = -1, xx = = aa. . We could do a limit We could do a limit calculation to find the calculation to find the derivative value derivative value but we but we will graph it on the GC and will graph it on the GC and graph its derivative.graph its derivative.So the derivative function So the derivative function equation is the “combined equation is the “combined derivative” f `(x) = 4xderivative” f `(x) = 4x33 – – 3x3x22 + 2x – ½ + 2x – ½

(N) Links(N) LinksVisual Calculus - Differentiation FormulasVisual Calculus - Differentiation Formulas

Calculus I (Math 2413) - Derivatives - DiffeCalculus I (Math 2413) - Derivatives - Differentiation Formulas from Paul Dawkinsrentiation Formulas from Paul Dawkins

Calc101.com Automatic Calculus featuring Calc101.com Automatic Calculus featuring a Differentiation Calculatora Differentiation Calculator

Some on-line questions with hints and soluSome on-line questions with hints and solutionstions

(O) Homework(O) HomeworkMCB4U – Nelson text, MCB4U – Nelson text, page 222, page 222, Q2efghij,3efghi,5cdehi,7,8,16,13Q2efghij,3efghi,5cdehi,7,8,16,13page 229, Q2,3,9,11,13,14page 229, Q2,3,9,11,13,14

IBHLY2 – Stewart, 1989, Chap 2.2, IBHLY2 – Stewart, 1989, Chap 2.2, p83, Q7-10p83, Q7-10Stewart, 1989, Chap 2.3, p88, Q1-Stewart, 1989, Chap 2.3, p88, Q1-3eol,6-103eol,6-10