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derivative

# derivative - The derivative as a function Section 3.2...

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Section 3.2 The derivative as a function

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Slope of the tangent line to the graph at the point (a,f(a)) f’(a) = lim h ! 0 f ( a + h ) " f ( a ) h = lim x ! a f ( x ) " f ( a ) x " a DERIVATIVE at A POINT a when it exists Instantaneous rate of change of f at a h=x-a
Differentiability at a f is differentiable at a if f’(a) exists • f should be continuous at a • the graph should have a tangent line at (a,f(a)) • this tangent line should not be vertical corner discontinuity vertical tangent 3 points in which f is not differentiable f’(a) = Slope of the tangent line to the graph at the point (a,f(a))

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f(x)= 1/(x-1) is not differentiable at 1 not continuous at 1 no tangent line at -1 f(x)= |x+1| is not differentiable at -1 f(x)= x 2/3 is not differentiable at 0 vertical tangent at 0
Continuous at -1 but not differentiable DIFFERENTIABLE CONTINUOUS CONTINUOUS DIFFERENTIABLE f(x)=|x+1|

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lim h ! 0 f ( a + h ) " f ( a ) h DERIVATIVE as A FUNCTION OF X = Slope of the tangent line to the graph at the point (a,f(a)) f’(a) = = Instantaneous rate of change of f at a For all a where f is differentiable we defined the derivative This gives a function of x (with domain = all differentiability points)
Graph of f(x) Graph f’(x)

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derivative - The derivative as a function Section 3.2...

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