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## The Derivative

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**The Derivative**3.1**Calculus**Differential Calc Integral Calc**Derivative – instantaneous rate of change of one variable**wrt another. Differentiation – process of finding the derivative**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: 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!**y**(b, f(b)) (a, f(a)) a b x Instantaneous Rate of Change We take the limit of the average rate of change as we let the intervals get smaller and smaller ∆x 0 Tangent Line**Tangent Line**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.**The derivative is same as the slope of the tangent line**Derivative at any point To Find slope of tangent line at a given point. Plug given point in f’(x)**“the derivative of f with respect to x”**“f prime x” or “y prime” “the derivative of y with respect to x” or “dee why dee ecks” “the derivative of f with respect to x” or “dee eff dee ecks” “the derivative of f of x” “dee dee ecks uv eff uv ecks” or**The derivative is the slope of the original function.**The derivative is defined at the end points of a function on a closed interval.**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)