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

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 .

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 .

• 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  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  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) = 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

(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

(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?

(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

(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

• 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