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The Derivative 3.1

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Calculus

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Derivative – instantaneous rate of change of one variable wrt another. Differentiation – process of finding the derivative

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Finding Average Rate of Change A piece of chocolate is pulled from a refrigerator (6° C) and placed on a counter (22° C). The temperature of the chocolate is given by: Min Temp What is the average rate of change in the temperature of the chocolate from 8 to 20 minutes? The rate of change was not constant thru out the process. This only tell us what happened on average over a period of time!

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We take the limit of the average rate of change as we let the intervals get smaller and smaller ∆x 0 y (a, f(a)) (b, f(b)) xab Tangent Line

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Definition of Tangent Line The tangent line to the graph of y = f(x) at x = c is the line through the point (c, f(c)) with slope provided this limit exists. If the instantaneous rate of change of f(x) with respect to x exists at a point c, then it is the slope of the tangent line at that point.

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Find the slope of the tangent line at x = 3 if

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Derivative at any point To Find slope of tangent line at a given point. Plug given point in f’(x) The derivative is same as the slope of the tangent line

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“f prime x”or “the derivative of f with respect to x” “y prime” “dee why dee ecks” or “the derivative of y with respect to x” “dee eff dee ecks” or “the derivative of f with respect to x” “dee dee ecks uv eff uv ecks”or “the derivative of f of x”

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The derivative is the slope of the original function. The derivative is defined at the end points of a function on a closed interval.

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Find the derivative of f(x) and use it to find the equation of the tangent line at the point x = 4 Slope: -19 and point (4, -28)

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Find dy/dx for

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