chapter5.pdf - M ATH 154 C HAPTER 5 Chapter 5 Differentiation 5.1 Computing Derivatives \u2022 Derivatives of power functions \u2022 Sums products and

# chapter5.pdf - M ATH 154 C HAPTER 5 Chapter 5...

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M ATH 154 C HAPTER 5 Chapter 5 Differentiation 5.1 Computing Derivatives Derivatives of power functions Sums, products, and quotients 5.2 Higher derivatives and antiderivatives Higher derivatives Antiderivatives 5.3 Position, velocity, and acceleration Goal: Learn to efficiently compute derivatives and antiderivatives. Application: We use our knowledge of antiderivatives to establish the basic principles of position, velocity, and acceleration. 1
M ATH 154 C HAPTER 5 5.1 Computing Derivatives Power functions Anytime n is a positive real number and x is a fixed real number, we have ( x + h ) n - x n = nx n - 1 h + h 2 g (h)Wheregis a polynomial function ofh.Let’s use the idea to compute thederivative ofxn. 2
M ATH 154 C HAPTER 5 Constant multiples Question.Supposefis a function andcis a constant. The functioncfisthen a constant multiple off, what is its derivative? Problem.Ify= 10x7what isy0? 3
M ATH 154 C HAPTER 5 Sums Question.Iffandgare functions, what is the derivative off+g?(f+g)0(x) = limh0(f+g)(x+h)-(f+g)(x)h= limh0f(x+h) +g(x+h)-f(x)-g(x)h= limh0f(x+h)-f(x)h+g(x+h)-g(x)h=f0(x) +g0(x)Sum rule.(f+g)0=f0+g0Using the sum rule and our existing knowledge about differentiating powerfunctions we can now differentiate any polynomial function.Problem.Compute the derivative ofy= 3x10+ 7x7-3x5-4x2+ 11x+ 3 4
M ATH 154 C HAPTER 5 Products