Math 151, Spring 2010,cBenjamin Aurispa3.1 DerivativesDefinition of the Derivative (at a point): The derivative of a function f at a number a, denoted f′(a), isf′(a) = limh→0f (a + h) − f (a)hor equivalently f′(a) = limx→af (x) − f (a)x − aWe have already seen all of this in the previous section! The derivative of a fun ction at a has the followinginterpretations:• f′(a) is the slope of the tangent line to the graph of f at x = a.• f′(a) is the instantaneous rate of change of the function f at x = a.• If f (t) is a position function, then f′(a) is the instantaneous velocity at time t = a.Either of the above formulas can be used to find the derivative of a function at a specific value x = a. Wedid m any examples of this in the previous section. We can also find a function that will give the derivativeat all values of x for which it is defined.Definition of the Derivative (as a function): If f is a function, the derivative of f , denoted f′isf′(x) = limh→0f (x + h) − f (x)hExample: Find the derivative of f (x) =x + 12x − 11Math 151, Spring 2010,cBenjamin AurispaExample: Find f′if f (x) =1√x − 1.There are many different notations for the derivative of a function. If y = f (x), the following notationsdenote the derivative of f :f′(x), y′,dydx,dfdx,ddxf (x)Taking the derivative of a function is also referred to as differentiation. A function is differentiable ata if f′(a) exists.A function is not differentiable at:• Discontinuities• Corners• Vertical tangent lines2Math 151, Spring 2010,cBenjamin AurispaWhere is the following f unction not differentiable?2 4 6 8 10 14 16−2−6 −4426−2−4−6Example: Show why the function f (x) = |x − 2| is not differentiable at x = 2.3Math 151, Spring 2010,cBenjamin AurispaGiven each graph of f below, s ketch a graph of f′.2 4 6 8 10 12 14246−2−42 4 6 8 10 12 14242 4 6 8 10 12 14246−2−42 4 6 8 10 12
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