(2) g(x)) = f(z) g(z) + g(z) - f'(x). (a) For each of the following functions, use the Product Rule to find the function's derivative. Notice the label of the derivative (e.g., the derivative of g(z) should be labeled g/(z)). (i) If g(z) = zsin(z), then g/(z) = (ii) If h(x)=ze", then (iii) If p(x)=In(az), then (iv.) If q(z)=z³ cos(z), then (v.) If r(a) e sin(x), then (b) Use your work in (a) to help you evaluate the following indefinite integrals. Use differentiation to check your work. (Don't forget the "+C".) (i) ze² + e' dz= (ii) fe* (sin(z) + cos(z)) dx = (iii) 2x cos(z) - zª sin(x) dx = (iv) [z cos(z) + sin(z) dx = (v) [1+In(z) dz= (c) Observe that the examples in (b) work nicely because of the derivatives you were asked to calculate in (a). Each integrand in (b) is precisely the result of differentiating one of the products of basic functions found in (a). To see what happens when an integrand is still a product but not necessarily the result of differentiating an elementary product, we consider how to evaluate [z cos(z) dz. (i) First, observe that zsin(x)) = cos(x)+sin()

(2) g(x)) = f(z) g(z) + g(z) - f'(x). (a) For each of the following functions, use the Product Rule to find the function's derivative. Notice the label of the derivative (e.g., the derivative of g(z) should be labeled g/(z)). (i) If g(z) = zsin(z), then g/(z) = (ii) If h(x)=ze", then (iii) If p(x)=In(az), then (iv.) If q(z)=z³ cos(z), then (v.) If r(a) e sin(x), then (b) Use your work in (a) to help you evaluate the following indefinite integrals. Use differentiation to check your work. (Don't forget the "+C".) (i) ze² + e' dz= (ii) fe* (sin(z) + cos(z)) dx = (iii) 2x cos(z) - zª sin(x) dx = (iv) [z cos(z) + sin(z) dx = (v) [1+In(z) dz= (c) Observe that the examples in (b) work nicely because of the derivatives you were asked to calculate in (a). Each integrand in (b) is precisely the result of differentiating one of the products of basic functions found in (a). To see what happens when an integrand is still a product but not necessarily the result of differentiating an elementary product, we consider how to evaluate [z cos(z) dz. (i) First, observe that zsin(x)) = cos(x)+sin()

Calculus For The Life Sciences

2nd Edition

ISBN:9780321964038

Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Chapter4: Calculating The Derivative

Section4.2: Derivatives Of Products And Quotients

Problem 35E

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