Let [tex]f(x)=3 \sec (4 x)[/tex]
[tex]f^{\prime}(x)=[/tex]
Answer (1)
Apply the chain rule to the function f ( x ) = 3 sec ( 4 x ) .
Recall that the derivative of sec ( u ) is sec ( u ) tan ( u ) ⋅ u ′ , where u = 4 x .
Calculate the derivative: f ′ ( x ) = 3 ⋅ sec ( 4 x ) tan ( 4 x ) ⋅ 4 .
Simplify to obtain the final answer: 12 sec ( 4 x ) tan ( 4 x ) .
Explanation
Problem Analysis We are given the function f ( x ) = 3 sec ( 4 x ) and we need to find its derivative, f ′ ( x ) .
Applying the Chain Rule To find the derivative, we will use the chain rule. Recall that the derivative of sec ( u ) is sec ( u ) tan ( u ) ⋅ u ′ , where u ′ is the derivative of u with respect to x . In our case, u = 4 x .
Differentiating So, we have: f ′ ( x ) = 3 ⋅ d x d [ sec ( 4 x )] Applying the chain rule, we get: f ′ ( x ) = 3 ⋅ sec ( 4 x ) tan ( 4 x ) ⋅ d x d [ 4 x ]
Calculating the Derivative of 4x The derivative of 4 x with respect to x is simply 4. Therefore, f ′ ( x ) = 3 ⋅ sec ( 4 x ) tan ( 4 x ) ⋅ 4
Simplifying the Expression Now, we simplify the expression: f ′ ( x ) = 12 sec ( 4 x ) tan ( 4 x )
Final Answer Thus, the derivative of f ( x ) = 3 sec ( 4 x ) is f ′ ( x ) = 12 sec ( 4 x ) tan ( 4 x ) .
Examples
In physics, understanding the derivatives of trigonometric functions like secant is crucial when analyzing oscillatory motion or wave phenomena. For instance, if f ( x ) = 3 sec ( 4 x ) represents the amplitude of a wave at a certain point, then f ′ ( x ) = 12 sec ( 4 x ) tan ( 4 x ) describes how the amplitude changes over time or space. This is essential for predicting wave behavior and designing systems that interact with waves, such as antennas or acoustic devices.
