Differentiate. [tex]F(x)=\frac{e^{3 x}}{x^6}[/tex]
Answer (1)
Apply the quotient rule: F ′ ( x ) = [ v ( x ) ] 2 u ′ ( x ) v ( x ) − u ( x ) v ′ ( x ) , where u ( x ) = e 3 x and v ( x ) = x 6 .
Find the derivatives: u ′ ( x ) = 3 e 3 x and v ′ ( x ) = 6 x 5 .
Substitute into the quotient rule: F ′ ( x ) = ( x 6 ) 2 ( 3 e 3 x ) ( x 6 ) − ( e 3 x ) ( 6 x 5 ) .
Simplify the expression: F ′ ( x ) = x 7 3 e 3 x ( x − 2 ) .
F ′ ( x ) = x 7 3 e 3 x ( x − 2 )
Explanation
Problem Analysis and Strategy We are given the function F ( x ) = x 6 e 3 x and we want to find its derivative, F ′ ( x ) . To do this, we will use the quotient rule. The quotient rule states that if we have a function of the form F ( x ) = v ( x ) u ( x ) , then the derivative is given by:
F ′ ( x ) = [ v ( x ) ] 2 u ′ ( x ) v ( x ) − u ( x ) v ′ ( x )
Finding Derivatives of u(x) and v(x) In our case, we can identify u ( x ) = e 3 x and v ( x ) = x 6 . Now we need to find the derivatives of u ( x ) and v ( x ) .
The derivative of u ( x ) = e 3 x is found using the chain rule: u ′ ( x ) = 3 e 3 x .
The derivative of v ( x ) = x 6 is found using the power rule: v ′ ( x ) = 6 x 5 .
Applying the Quotient Rule Now we can plug these derivatives into the quotient rule formula:
F ′ ( x ) = ( x 6 ) 2 ( 3 e 3 x ) ( x 6 ) − ( e 3 x ) ( 6 x 5 )
Simplifying the Expression Next, we simplify the expression:
F ′ ( x ) = x 12 3 x 6 e 3 x − 6 x 5 e 3 x
We can factor out 3 x 5 e 3 x from the numerator:
F ′ ( x ) = x 12 3 x 5 e 3 x ( x − 2 )
Final Simplification Finally, we can simplify by canceling out x 5 from the numerator and denominator:
F ′ ( x ) = x 7 3 e 3 x ( x − 2 )
Final Answer Therefore, the derivative of F ( x ) = x 6 e 3 x is:
F ′ ( x ) = x 7 3 e 3 x ( x − 2 )
Examples
In physics, you might encounter this type of function when modeling the decay of a signal (represented by the exponential term) that is also affected by some time-dependent factor (represented by the polynomial term). Finding the derivative helps you understand how the rate of change of the signal evolves over time, which is crucial for analyzing the system's behavior.
