Leibniz: dy/dx (read "dy dx" )
1646­1716
Newton: y' (read "y prime" ) also: f '(x)
1643­1727
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3.1­Derivative of a function
Q = {
h
av ROC = slopesec = slope PQ = f(x+h) ­ f(x)
h
Instantaneous ROC = Derivative f '(x) = slopetan = lim f(x+h) ­ f(x)
h 0 h GSP
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y = f(x) sketch the derivative
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let's try to find the derivative of a function at a "point" with our calculator
1. f(x) = .5x2 find f '(x) when x = 3 :(Find: (f '(3))
do you already know the derivative of f?
(yes, f '(x) = 2*.5x2­1 = 1x by the "Power Rule shortcut"; so f '(3) = 1(3) = 3
calculator:
MATH #8: nDeriv(.5x2, x, 3) enter
(function, variable, numerical value)
explain your answer "geometrically"
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2. f(x) = sinx
use nDeriv to find the slope of the sine curve at x = π
solution: nDeriv(sinx, x , π)
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Can we use our calculator to create a graph of the derivative?
3. y1 = sinx
y2 = nDeriv(y1,x,x)
?? ­ comments? d(sinx) = cosx
4. y1 = ex
y2 = nDeriv(y1,x,x) ?? ­ comments? d(ex) = ex
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