Unit 2 Calculating Derivatives From first principles!

Necessary Skills Powers of Binomials How do you expand (a+b) 2 ? (a+b) 2 =(a+b)(a+b) (a+b) 2 =a 2 +ab+ab+b 2 (a+b) 2 =a 2 +2ab+b 2

(a+b) 3 Expand (a+b) 3 (a+b) 3 =(a+b)(a 2 +2ab+b 2 ) (a+b) 3 =a 3 +2a 2 b+ab 2 + a 2 b+2ab 2 +b 3 (a+b) 3 =a 3 +3a 2 b+3ab 2 +b 3 Pascals Triangle can make this easier

Pascal’s Triangle

Example 1 Expand (2 – 3x) 3 Let a = 2 and b = -3x (a+b) 3 =a 3 +3a 2 b+3ab 2 +b 3 So (2 – 3x) 3 =(2) 3 +3(2) 2 (-3x)+3(2)(-3x) 2 +(-3x) 3 (2 – 3x) 3 =8+3(4)(-3x)+6(9x 2 )+-27x 3 (2 – 3x) 3 =8 - 36x + 54x x 3 (2 – 3x) 3 = - 27x x x+8

You try Expand (3a+2b) 4

Homework Page 72 #1 a) – g), 2

2.1 The Derivative as a Limit See the gsp demo demodemo Let P be any point on the graph of the function y=f(x). P is P(x, f(x)) Let Q be any point on the graph near P. Q is Q(x+h,f(x+h)) h is some small number.

Slope of the secant The slope of the secant PQ is: As h gets smaller and smaller what happens to Q? It gets closer to P. What happens to the secant PQ as h gets close to zero? The secant comes closer to the tangent. What happens to the slope of the secant PQ? It gets closer to the slope of the tangent.

0/0= indeterminate As Q approaches P what value does h approach? h approaches the value of zero. Calculate the slope when h=0. This is not defined, but we can use the formula with what is called a limit.

y=x 2 Let P(x,x 2 ) be any point on f(x)=x 2 Let Q(x+h, (x+h) 2 ) be a point on the graph near P. The slope of PQ is: Factor Expand Let h get smaller and smaller. The limit as h approaches zero Divide by h We can’t substitute h=0. Why?

First Principles Definition of the Derivative When h approaches 0, the slope of PQ approaches: The derivative of a function f is a new function f ‘ (f prime) As h approaches 0, (2x+h) approaches 2x. The slope of PQ approaches the slope of the tangent. The slope of the tangent is 2x. That is the derivative is the limit of the difference quotient (limiting slope of the secant)

Using first principles Use first principles to determine the derivative of f(x)=3x 2 +2x Sub x+h Expand Collect like terms

Example continued Factor out h Divide out h

You try a) Use the first principles definition to differentiate the function y= 6x - x 2 b) Sketch the function in part a) and its derivative. c) Determine the equation of the tangent line at the point (1,5) on the graph. Draw the tangent on the graph of the function.

You try – solution.

Homework Pg 78 –80 #2,5,6,8,10 – 16