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 √ 2 g(x) = xex √ 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 √ 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 √ 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 √ 2 g(x) = xex √ 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 √ 2 g(x) = xex √ 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