The Product Rule Math 130 - Essentials of Calculus 5 March 2021 Math 130 - Essentials of Calculus The Product Rule 5 March 2021 1/5 Unfortunately, it is not as simple d as [f (x)g(x)] = f 0(x)g0(x)... dx Theorem (The Product Rule) If f and g are both differentiable, then d [f (x)g(x)] = f 0(x)g(x) + f (x)g0(x): dx Section 3.3 - The Product and Quotient Rules Expanding Our Abilities So far, we’ve learned the power rule and how to differentiate functions of the form cf (x) and f (x) ± g(x). How would we differentiate f (x)g(x)? Math 130 - Essentials of Calculus The Product Rule 5 March 2021 2/5 Theorem (The Product Rule) If f and g are both differentiable, then d [f (x)g(x)] = f 0(x)g(x) + f (x)g0(x): dx Section 3.3 - The Product and Quotient Rules Expanding Our Abilities So far, we’ve learned the power rule and how to differentiate functions of the form cf (x) and f (x) ± g(x). How would we differentiate f (x)g(x)? Unfortunately, it is not as simple d as [f (x)g(x)] = f 0(x)g0(x)... dx Math 130 - Essentials of Calculus The Product Rule 5 March 2021 2/5 Section 3.3 - The Product and Quotient Rules Expanding Our Abilities So far, we’ve learned the power rule and how to differentiate functions of the form cf (x) and f (x) ± g(x). How would we differentiate f (x)g(x)? Unfortunately, it is not as simple d as [f (x)g(x)] = f 0(x)g0(x)... dx Theorem (The Product Rule) If f and g are both differentiable, then d [f (x)g(x)] = f 0(x)g(x) + f (x)g0(x): dx Math 130 - Essentials of Calculus The Product Rule 5 March 2021 2/5 p 2 g(x) = xex p 3 R(t) = (t + et )(3 − t) 4 k(x) = sin x cos x Section 3.3 - The Product and Quotient Rules Examples Example Differentiate the following functions 1 f (x) = xex Math 130 - Essentials of Calculus The Product Rule 5 March 2021 3/5 p 3 R(t) = (t + et )(3 − t) 4 k(x) = sin x cos x Section 3.3 - The Product and Quotient Rules Examples Example Differentiate the following functions 1 f (x) = xex p 2 g(x) = xex Math 130 - Essentials of Calculus The Product Rule 5 March 2021 3/5 4 k(x) = sin x cos x Section 3.3 - The Product and Quotient Rules Examples Example Differentiate the following functions 1 f (x) = xex p 2 g(x) = xex p 3 R(t) = (t + et )(3 − t) Math 130 - Essentials of Calculus The Product Rule 5 March 2021 3/5 Section 3.3 - The Product and Quotient Rules Examples Example Differentiate the following functions 1 f (x) = xex p 2 g(x) = xex p 3 R(t) = (t + et )(3 − t) 4 k(x) = sin x cos x Math 130 - Essentials of Calculus The Product Rule 5 March 2021 3/5 Example Find the second derivative of f (x) = x3 sin x. Section 3.3 - The Product and Quotient Rules More Examples Example Find the derivative of f (x) = x2ex cos x. Math 130 - Essentials of Calculus The Product Rule 5 March 2021 4/5 Section 3.3 - The Product and Quotient Rules More Examples Example Find the derivative of f (x) = x2ex cos x. Example Find the second derivative of f (x) = x3 sin x. Math 130 - Essentials of Calculus The Product Rule 5 March 2021 4/5 Property (2nd Product Rule) If f and g are twice differentiable functions, then d 2 [f (x)g(x)] = f 00(x)g(x) + 2f 0(x)g0(x) + f (x)g00(x): dx2 Section 3.3 - The Product and Quotient Rules Fun Facts Property (Triple Product Rule) If f , g, and h are differentiable functions, then d [f (x)g(x)h(x)] = f 0(x)g(x)h(x) + f (x)g0(x)h(x) + f (x)g(x)h0(x): dx Math 130 - Essentials of Calculus The Product Rule 5 March 2021 5/5 Section 3.3 - The Product and Quotient Rules Fun Facts Property (Triple Product Rule) If f , g, and h are differentiable functions, then d [f (x)g(x)h(x)] = f 0(x)g(x)h(x) + f (x)g0(x)h(x) + f (x)g(x)h0(x): dx Property (2nd Product Rule) If f and g are twice differentiable functions, then d 2 [f (x)g(x)] = f 00(x)g(x) + 2f 0(x)g0(x) + f (x)g00(x): dx2 Math 130 - Essentials of Calculus The Product Rule 5 March 2021 5/5.