The product rule for differentation E. Kim 1 Product Rule for - PowerPoint PPT Presentation

The product rule for differentation E. Kim 1

Product Rule for Differentiation Goal Starting with differentiable functions f ( x ) and g ( x ) , we want to get the derivative of f ( x ) g ( x ) . 2

Product Rule for Differentiation Goal Starting with differentiable functions f ( x ) and g ( x ) , we want to get the derivative of f ( x ) g ( x ) . Previously, we saw [ f ( x ) + g ( x )] ′ = f ′ ( x ) + g ′ ( x ) “Sum Rule” 2

Product Rule for Differentiation Goal Starting with differentiable functions f ( x ) and g ( x ) , we want to get the derivative of f ( x ) g ( x ) . Previously, we saw [ f ( x ) + g ( x )] ′ = f ′ ( x ) + g ′ ( x ) “Sum Rule” Question Is the Product Rule [ f ( x ) g ( x )] ′ = f ′ ( x ) g ′ ( x ) or not? 2

Is [ f ( x ) g ( x )] ′ = f ′ ( x ) g ′ ( x ) the Product Rule? 3

Is [ f ( x ) g ( x )] ′ = f ′ ( x ) g ′ ( x ) the Product Rule? Let’s test it out! Choose: 3

Is [ f ( x ) g ( x )] ′ = f ′ ( x ) g ′ ( x ) the Product Rule? Let’s test it out! Choose: f ( x ) = x 3 3

Is [ f ( x ) g ( x )] ′ = f ′ ( x ) g ′ ( x ) the Product Rule? Let’s test it out! Choose: f ( x ) = x 3 g ( x ) = x 10 3

Is [ f ( x ) g ( x )] ′ = f ′ ( x ) g ′ ( x ) the Product Rule? Let’s test it out! Choose: f ( x ) = x 3 g ( x ) = x 10 k ( x ) = f ( x ) · g ( x ) = x 13 3

Is [ f ( x ) g ( x )] ′ = f ′ ( x ) g ′ ( x ) the Product Rule? Let’s test it out! Choose: f ( x ) = x 3 g ( x ) = x 10 k ( x ) = f ( x ) · g ( x ) = x 13 Compute derivatives: 3

Is [ f ( x ) g ( x )] ′ = f ′ ( x ) g ′ ( x ) the Product Rule? Let’s test it out! Choose: f ( x ) = x 3 g ( x ) = x 10 k ( x ) = f ( x ) · g ( x ) = x 13 Compute derivatives: f ′ ( x ) = 3

Is [ f ( x ) g ( x )] ′ = f ′ ( x ) g ′ ( x ) the Product Rule? Let’s test it out! Choose: f ( x ) = x 3 g ( x ) = x 10 k ( x ) = f ( x ) · g ( x ) = x 13 Compute derivatives: f ′ ( x ) = 3 x 2 3

Is [ f ( x ) g ( x )] ′ = f ′ ( x ) g ′ ( x ) the Product Rule? Let’s test it out! Choose: f ( x ) = x 3 g ( x ) = x 10 k ( x ) = f ( x ) · g ( x ) = x 13 Compute derivatives: f ′ ( x ) = 3 x 2 g ′ ( x ) = 3

Is [ f ( x ) g ( x )] ′ = f ′ ( x ) g ′ ( x ) the Product Rule? Let’s test it out! Choose: f ( x ) = x 3 g ( x ) = x 10 k ( x ) = f ( x ) · g ( x ) = x 13 Compute derivatives: f ′ ( x ) = 3 x 2 g ′ ( x ) = 10 x 9 3

Is [ f ( x ) g ( x )] ′ = f ′ ( x ) g ′ ( x ) the Product Rule? Let’s test it out! Choose: f ( x ) = x 3 g ( x ) = x 10 k ( x ) = f ( x ) · g ( x ) = x 13 Compute derivatives: k ′ ( x ) = [ f ( x ) g ( x )] ′ = f ′ ( x ) = 3 x 2 g ′ ( x ) = 10 x 9 3

Is [ f ( x ) g ( x )] ′ = f ′ ( x ) g ′ ( x ) the Product Rule? Let’s test it out! Choose: f ( x ) = x 3 g ( x ) = x 10 k ( x ) = f ( x ) · g ( x ) = x 13 Compute derivatives: k ′ ( x ) = [ f ( x ) g ( x )] ′ = 13 x 12 f ′ ( x ) = 3 x 2 g ′ ( x ) = 10 x 9 3

Is [ f ( x ) g ( x )] ′ = f ′ ( x ) g ′ ( x ) the Product Rule? Let’s test it out! Choose: f ( x ) = x 3 g ( x ) = x 10 k ( x ) = f ( x ) · g ( x ) = x 13 Compute derivatives: k ′ ( x ) = [ f ( x ) g ( x )] ′ = 13 x 12 f ′ ( x ) = 3 x 2 g ′ ( x ) = 10 x 9 Compare: 3

Is [ f ( x ) g ( x )] ′ = f ′ ( x ) g ′ ( x ) the Product Rule? Let’s test it out! Choose: f ( x ) = x 3 g ( x ) = x 10 k ( x ) = f ( x ) · g ( x ) = x 13 Compute derivatives: k ′ ( x ) = [ f ( x ) g ( x )] ′ = 13 x 12 f ′ ( x ) = 3 x 2 g ′ ( x ) = 10 x 9 Compare: ◮ [ f ( x ) g ( x )] ′ = 13 x 12 3

Is [ f ( x ) g ( x )] ′ = f ′ ( x ) g ′ ( x ) the Product Rule? Let’s test it out! Choose: f ( x ) = x 3 g ( x ) = x 10 k ( x ) = f ( x ) · g ( x ) = x 13 Compute derivatives: k ′ ( x ) = [ f ( x ) g ( x )] ′ = 13 x 12 f ′ ( x ) = 3 x 2 g ′ ( x ) = 10 x 9 Compare: ◮ [ f ( x ) g ( x )] ′ = 13 x 12 ◮ f ′ ( x ) g ′ ( x ) = (3 x 2 )(10 x 9 ) = 30 x 11 3

Is [ f ( x ) g ( x )] ′ = f ′ ( x ) g ′ ( x ) the Product Rule? Let’s test it out! Choose: f ( x ) = x 3 g ( x ) = x 10 k ( x ) = f ( x ) · g ( x ) = x 13 Compute derivatives: k ′ ( x ) = [ f ( x ) g ( x )] ′ = 13 x 12 f ′ ( x ) = 3 x 2 g ′ ( x ) = 10 x 9 Compare: ◮ [ f ( x ) g ( x )] ′ = 13 x 12 ◮ f ′ ( x ) g ′ ( x ) = (3 x 2 )(10 x 9 ) = 30 x 11 No! This is NOT the Product Rule! [ f ( x ) g ( x )] ′ � = f ′ ( x ) g ′ ( x ) 3

Then what is the Product Rule? Intuitively... it’s like working at Culver’s Say you worked at Culver’s at a rate of r = 7 . 75 per hour for h = 20 hours each week. Your take-home pay is p = rh . How can your take-home pay go up? Culver’s in Onalaska, WI Source: Wikipedia 4

Then what is the Product Rule? Intuitively... it’s like working at Culver’s Say you worked at Culver’s at a rate of r = 7 . 75 per hour for h = 20 hours each week. Your take-home pay is p = rh . How can your take-home pay go up? ◮ Pay rate goes up: r � r new r new = r + ∆ r Culver’s in Onalaska, WI Source: Wikipedia 4

Then what is the Product Rule? Intuitively... it’s like working at Culver’s Say you worked at Culver’s at a rate of r = 7 . 75 per hour for h = 20 hours each week. Your take-home pay is p = rh . How can your take-home pay go up? ◮ Pay rate goes up: r � r new r new = r + ∆ r Culver’s in Onalaska, WI ◮ Hours per week goes up: h � h new Source: Wikipedia h new = h + ∆ h 4

Then what is the Product Rule? Intuitively... it’s like working at Culver’s Say you worked at Culver’s at a rate of r = 7 . 75 per hour for h = 20 hours each week. Your take-home pay is p = rh . How can your take-home pay go up? ◮ Pay rate goes up: r � r new r new = r + ∆ r Culver’s in Onalaska, WI ◮ Hours per week goes up: h � h new Source: Wikipedia h new = h + ∆ h ◮ Both r and h increase 4

Then what is the Product Rule? Intuitively... it’s like working at Culver’s Say you worked at Culver’s at a rate of r = 7 . 75 per hour for h = 20 hours each week. Your take-home pay is p = rh . How can your take-home pay go up? ◮ Pay rate goes up: r � r new r new = r + ∆ r Culver’s in Onalaska, WI ◮ Hours per week goes up: h � h new Source: Wikipedia h new = h + ∆ h ◮ Both r and h increase ◮ p old = rh 4

Then what is the Product Rule? Intuitively... it’s like working at Culver’s Say you worked at Culver’s at a rate of r = 7 . 75 per hour for h = 20 hours each week. Your take-home pay is p = rh . How can your take-home pay go up? ◮ Pay rate goes up: r � r new r new = r + ∆ r Culver’s in Onalaska, WI ◮ Hours per week goes up: h � h new Source: Wikipedia h new = h + ∆ h ◮ Both r and h increase ◮ p old = rh ◮ p new = r new h new = ( r + ∆ r )( h + ∆ h ) 4

Then what is the Product Rule? Intuitively... it’s like working at Culver’s Say you worked at Culver’s at a rate of r = 7 . 75 per hour for h = 20 hours each week. Your take-home pay is p = rh . How can your take-home pay go up? ◮ Pay rate goes up: r � r new r new = r + ∆ r Culver’s in Onalaska, WI ◮ Hours per week goes up: h � h new Source: Wikipedia h new = h + ∆ h ◮ Both r and h increase ◮ p old = rh ◮ p new = r new h new = ( r + ∆ r )( h + ∆ h ) ◮ Change in pay ∆ p = p new − p old = ( r + ∆ r )( h + ∆ h ) − rh 4

Then what is the Product Rule? Intuitively... it’s like working at Culver’s Say you worked at Culver’s at a rate of r = 7 . 75 per hour for h = 20 hours each week. Your take-home pay is p = rh . How can your take-home pay go up? ◮ Pay rate goes up: r � r new r new = r + ∆ r Culver’s in Onalaska, WI ◮ Hours per week goes up: h � h new Source: Wikipedia h new = h + ∆ h ◮ Both r and h increase ◮ p old = rh ◮ p new = r new h new = ( r + ∆ r )( h + ∆ h ) ◮ Change in pay ∆ p = p new − p old = ( r + ∆ r )( h + ∆ h ) − rh = ( rh + r ∆ h + h ∆ r + ∆ r ∆ h ) − rh 4

Intuitive idea of the Product Rule ∆ r h ∆ r ∆ r ∆ h r r ∆ h rh h ∆ h ∆ p = ( rh + r ∆ h + h ∆ r + ∆ r ∆ h ) − rh 5

Intuitive idea of the Product Rule ∆ r h ∆ r ∆ r ∆ h r r ∆ h rh h ∆ h ∆ p = ( rh + r ∆ h + h ∆ r + ∆ r ∆ h ) − rh 5

Intuitive idea of the Product Rule ∆ r h ∆ r ∆ r ∆ h r r ∆ h rh h ∆ h ∆ p = ( rh + r ∆ h + h ∆ r + ∆ r ∆ h ) − rh ∆ p = r ∆ h + h ∆ r + ∆ r ∆ h � �� � 5

Intuitive idea of the Product Rule ∆ r h ∆ r ∆ r ∆ h r r ∆ h rh h ∆ h ∆ p = ( rh + r ∆ h + h ∆ r + ∆ r ∆ h ) − rh ∆ p = r ∆ h + h ∆ r + ∆ r ∆ h � �� � negligible 5

Intuitive idea of the Product Rule ∆ r h ∆ r ∆ r ∆ h r r ∆ h rh h ∆ h ∆ p = ( rh + r ∆ h + h ∆ r + ∆ r ∆ h ) − rh ∆ p = r ∆ h + h ∆ r + ∆ r ∆ h � �� � negligible ∆ p ≈ r ∆ h + h ∆ r The change in the product p = rh is the old rate r times the change in hours (∆ h ) , plus the old hours h times the change in rate (∆ r ) . 5

Deriving the Product Rule Starting with differentiable functions f ( x ) and g ( x ) , we want to get the derivative of f ( x ) g ( x ) . 6