If [tex]$f(x)=\frac{(10 x+9)^4}{8 x+5}$[/tex], what is [tex]$f^{\prime}(x)$[/tex]?
Answer (1)
Apply the quotient rule: f ′ ( x ) = [ v ( x ) ] 2 u ′ ( x ) v ( x ) − u ( x ) v ′ ( x ) , where u ( x ) = ( 10 x + 9 ) 4 and v ( x ) = 8 x + 5 .
Find the derivatives: u ′ ( x ) = 40 ( 10 x + 9 ) 3 and v ′ ( x ) = 8 .
Substitute into the quotient rule: f ′ ( x ) = ( 8 x + 5 ) 2 40 ( 10 x + 9 ) 3 ( 8 x + 5 ) − ( 10 x + 9 ) 4 ( 8 ) .
Simplify the expression: f ′ ( x ) = ( 8 x + 5 ) 2 ( 10 x + 9 ) 3 ( 240 x + 128 ) .
f ′ ( x ) = ( 8 x + 5 ) 2 ( 10 x + 9 ) 3 ( 240 x + 128 )
Explanation
Problem Analysis We are given the function f ( x ) = 8 x + 5 ( 10 x + 9 ) 4 and asked to find its derivative f ′ ( x ) . This requires the application of the quotient rule.
Quotient Rule The quotient rule states that if f ( x ) = v ( x ) u ( x ) , then f ′ ( x ) = [ v ( x ) ] 2 u ′ ( x ) v ( x ) − u ( x ) v ′ ( x ) . In our case, u ( x ) = ( 10 x + 9 ) 4 and v ( x ) = 8 x + 5 .
Derivatives of u(x) and v(x) Now we need to find the derivatives of u ( x ) and v ( x ) .
For u ( x ) = ( 10 x + 9 ) 4 , we use the chain rule: u ′ ( x ) = 4 ( 10 x + 9 ) 3 ⋅ 10 = 40 ( 10 x + 9 ) 3 .
For v ( x ) = 8 x + 5 , the derivative is simply v ′ ( x ) = 8 .
Applying the Quotient Rule Formula Now we substitute u ( x ) , v ( x ) , u ′ ( x ) , and v ′ ( x ) into the quotient rule formula:
f ′ ( x ) = ( 8 x + 5 ) 2 40 ( 10 x + 9 ) 3 ( 8 x + 5 ) − ( 10 x + 9 ) 4 ( 8 )
Simplifying the Expression Next, we simplify the expression:
f ′ ( x ) = ( 8 x + 5 ) 2 ( 10 x + 9 ) 3 [ 40 ( 8 x + 5 ) − 8 ( 10 x + 9 )]
Further Simplification Further simplification:
f ′ ( x ) = ( 8 x + 5 ) 2 ( 10 x + 9 ) 3 [ 320 x + 200 − 80 x − 72 ]
Final Simplification Final simplification:
f ′ ( x ) = ( 8 x + 5 ) 2 ( 10 x + 9 ) 3 ( 240 x + 128 )
Final Answer Therefore, the derivative of the given function is:
f ′ ( x ) = ( 8 x + 5 ) 2 ( 10 x + 9 ) 3 ( 240 x + 128 )
Examples
Understanding derivatives is crucial in many real-world applications. For instance, if f ( x ) represents the cost of producing x items, then f ′ ( x ) gives the marginal cost, which is the approximate cost of producing one more item. In economics and business, this helps in making decisions about production levels and pricing strategies. Similarly, in physics, if f ( x ) represents the position of an object at time x , then f ′ ( x ) gives the velocity of the object at time x .
