2Section Outline The Derivative Differentiation Slope and the DerivativeEquation of the Tangent Line to the Graph of y = f (x) at (a, f (a))Leibniz Notation for DerivativesCalculating Derivatives Via the Difference QuotientDifferentiableLimit Definition of the DerivativeLimit Calculation of the Derivative
3The Derivative Definition Example Derivative: The slope formula for a function y = f (x), denoted:Given the function f (x) = x3, the derivative is
4Differentiation Definition Example Differentiation: The process of computing a derivative.No example will be given at this time since we do not yet know how to compute derivatives. But don’t worry, you’ll soon be able to do basic differentiation in your sleep.
5Differentiation Examples These examples can be summarized by the following rule.
6Differentiation Examples Find the derivative ofSOLUTIONThis is the given equation.Rewrite the denominator as an exponent.Rewrite with a negative exponent.What we’ve done so far has been done for the sole purpose of rewriting the function in the form of f (x) = xr.Use the Power Rule where r = -1/7 and then simplify.
7Differentiation Examples Find the slope of the curve y = x5 at x = -2.SOLUTIONWe must first find the derivative of the given function.This is the given function.Use the Power Rule.Since the derivative function yields information about the slope of the original function, we can now use to determine the slope of the original function at x = -2.Replace x with -2.Evaluate.Therefore, the slope of the original function at x = -2 is 80 (or 80/1).
8Equation of the Tangent Line to the Graph of y = f (x) at the point (a, f (a))
9Equation of the Tangent Line EXAMPLEFind the equation of the tangent line to the graph of f (x) = 3x at x = 4.SOLUTIONWe must first find the derivative of the given function.This is the given function.Differentiate.Notice that in this case the derivative function is a constant function, 3. Therefore, at x = 4, or any other value, the value of the derivative will be 3. So now we use the Equation of the Tangent Line that we just saw.This is the Equation of the Tangent Line.f (4) = 12 andSimplify.
10Leibniz Notation for Derivatives Ultimately, this notation is a better and more effective notation for working with derivatives.
11Calculating Derivatives Via the Difference Quotient The Difference Quotient is
12Calculating Derivatives Via the Difference Quotient EXAMPLEApply the three-step method to compute the derivative of the following function:SOLUTIONSTEP 1: We calculate the difference quotient and simplify as much as possible.This is the difference quotient.Evaluate f (x + h) and f (x).Simplify.Simplify.Simplify.
13Calculating Derivatives Via the Difference Quotient CONTINUEDFactor.Cancel and simplify.STEP 2: As h approaches zero, the expression -2x – h approaches -2x, and hence the difference quotient approaches -2x.STEP 3: Since the difference quotient approaches the derivative of f (x) = -x2 + 2 as h approaches zero, we conclude that
14Differentiable Definition Example Differentiable: A function f is differentiable at x ifapproaches some number as h approaches zero.The function f (x) = |x| is differentiable for all values of x except x = 0 since the graph of the function has no definite slope when x = 0 (f is nondifferentiable at x = 0) but does have a definite slope (1 or -1) for every other value of x.
16Limit Calculation of the Derivative EXAMPLEUsing limits, apply the three-step method to compute the derivative of the following function:SOLUTIONSTEP 1:This is the difference quotient.Evaluate f (x + h) and f (x).Simplify.Simplify.Simplify.
17Limit Calculation of the Derivative CONTINUEDFactor.Cancel and simplify.STEP 2: As h approaches 0, the expression -2x – h approaches -2xSTEP 3: Since the difference quotient approaches , we conclude that