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

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

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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

MCB4U - Santowski

(A) Review and Difference Rule

- 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

- 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)

(i) f(x) = 3 is called a constant function Differentiation Rules 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(ii) f(x) = 4x or f(x) = -½x or f(x) = mx Differentiation Rules 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(iii) f(x) = x Differentiation Rules2 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(iv) f(x) = x Differentiation Rules3 or ¼ x3 or ax3?

What would be the rate of change of this function at x = 6? x = -1, x = a.

So the derivative function equation is the quadratic function f `(x) = 3ax2

(F) Cubic Functions(v) f(x) = x Differentiation Rules4 or ¼ x4 or ax4?

What would be the rate of change of this function at x = 6? x = -1, x = a.

So the derivative function equation is the cubic function f `(x) = 4ax3

(G) Quartic Functions(H) Summary Differentiation Rules

- 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 Differentiation Rules

- 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 Differentiation Rules

- 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 Differentiation Rules

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

(vi) f(x) = x² + 4x - 1 Differentiation Rules

What would be the rate of change of this function at x = 6? x = -1, x = a.

So the derivative function equation is the “combined derivative” of f `(x) = 2x + 4

(L) Example #1(L) Example #1 Differentiation Rules

(vii) f(x) = x Differentiation Rules4 – 3x3 + x2 -½x + 3

What would be the rate of change of this function at x = 6? x = -1, x = a.

So the derivative function equation is the “combined derivative” f `(x) = 4x3 – 3x2 + 2x – ½

(M) Example 2(N) Links Differentiation Rules

- 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 Differentiation Rules

- 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