Business Calculus

Derivatives

1.5 – 1.7 Derivative Rules

Linear Rule:

Constant Rule:

Power Rule:

baxxf )( axf )(

kxf )( 0)( xf

nxxf )( 1)( nnxxf

Coefficient Rule: )(xkfy )(xfkdx

dy

Sum/Difference Rule:

)()( xgxfy )()( xgxfdx

dy

If a function does not look like one of the rules, we mustrewrite the function so that it does.

Applications

When asked for the rate at which a quantity is growing or declining between two values, this is an average rate of change.

When asked for the rate of change at a single value, use the derivative.

More derivative rules

)()()()()( tgtftgtftH

1.6 Product Rule: )()()( tgtftH

Product Rule: "derivative of the first times the second + the first times the derivative of the second"

2)(

)()()()()(

xg

xgxfxfxgxH

1.6 Quotient Rule:

)(

)()(

xg

xfxH

Quotient Rule: “denominator times the derivative of the numerator minus the numerator times the derivative of the denominator, all divided by the denominator squared”

1.7 The Chain Rule :

))(())(()( xgfxgfxH

)())(()( xgxgfxH

"The derivative of the outside times the derivative of the inside."

1.8 Higher Order Derivatives

The second derivative is found by taking the derivative ofthe first derivative.

Notation for the second derivative of y = f (x):

1. 2. 3. 4.

2

2

dx

yd )(xf )(xfdx

d

dx

dy

dx

d

Notation for the third derivative of y = f (x):

1. 2. 3. 4.

3

3

dx

yd )(xf )(xfdx

d

2

2

dx

yd

dx

d