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Warm-up 1. Find the derivative f ( x) x 4 x 3 2 2. Find the derivative at the following point. f ( x) x 1 at x=3 Table of Contents • 10. Section 3.2 The Derivative as a Function The Derivative as a Function • Essential Question – What rules of differentiation will make it easier to calculate derivatives? Notation for Derivative dy d f ( x) or y or or f ( x) dx dx ' ' • If derivative exists, we say it is differentiable Power Rule • Power Rule n n 1 f '( x ) n x • Bring down the exponent and subtract one from the exponent dy 3 Example ( x ) 3x 2 dx Example - dy Example dx dy 2 1/3 ( x 2/3 ) x dx 3 dy 1/ 2 1 1/ 2 1 x x x dx 2 2 x More notation • d 4 x dx x4 means find the derivative of when x = -2 x 2 2 more rules • Constant multiple Example - y=4x 2 (cf ) cf ' ' y ' 4(2 x) 8x • Sum and Difference ( f g) f g ' Example - f '(2 x 3x) 4 x 3 2 ' ' Differentiating a polynomial dp 5 3 2 Find if p t 6t t 16 dt 3 dp 5 2 = 3t 12t 0 dt 3 Derivative of ex d x x e e dx Example • Find the equation of the tangent line to the graph of f(x) = 3ex -5x2 at x=2 f ' ( x) 3e x 10 x f ' (2) 3e2 10(2) 2.17 f (2) 3e2 5(2)2 2.17 y 2.17 2.17( x 2) y 2.17 x 2.17 What information does the derivative at a point tell us? • Tells us whether the tangent line has a positive or negative slope • Tells us how steep the line is (the larger the derivative, the steeper the line) • Tells us if there is a turning point (slope is 0) Horizontal Tangents • Does y = x4 – 2x2 + 2 have any horizontal tangents? • First find the derivative, then set = 0 (because the slope of a horizontal line is 0) dy = 4 x3 4 x 0 dx 4 x( x 2 1) 0 4x 0 ( x2 1) 0 x0 x 1 Calculator example • Find the points where horizontal tangents occur. s(t ) 0.2t 0.7t 2t 5t 4 ds 0.8t 3 2.1x 2 4 x 5 0 dt On calculator, find zeros 4 3 2 t 1.862, 0.9484, 3.539 Plug these back into O.F. to get points Graphing f’(x) from f(x) • Find slope at each point • Make a new graph using same x points and the slope as the y point • If f is increasing, f ‘ will be positive (above the x axis) • If f has a turning point, f ‘ will be 0 • If f is decreasing, f ‘ will be negative (below the x axis) Graph example • Given the graph of f(x), which of A or B is the derivative? Differentiability • Differentiability implies continuity • If f is differentiable at x = c, then f is continuous at x = c • The opposite is not true • A function can be continuous at x = c, but not differentiable 4 times a derivative fails to exist • Corner • Cusp 4 times a derivative fails to exist • Vertical tangent • Discontinuity Local Linearity • A function that is differentiable closely resembles its own tangent line when viewed very closely • In other words, when zoomed in on a few times, a curve will look like a straight line. Example yx 2 y x Assignment • Pg. 139 #1-11 odd, 17-29 odd, 47-57 odd, 71-75 odd, 76-80 all, 83-87 odd