Let $f(x)=4 \sec (4 x)$
$f^{\prime}(x) =$
Answer (1)
Apply the chain rule to find the derivative of f ( x ) = 4 sec ( 4 x ) .
Recognize that the derivative of sec ( u ) is sec ( u ) tan ( u ) ⋅ u ′ , where u = 4 x .
Calculate the derivative of the inner function: d x d ( 4 x ) = 4 .
Simplify the expression to obtain the final answer: 16 sec ( 4 x ) tan ( 4 x ) .
Explanation
Problem Analysis We are given the function f ( x ) = 4 sec ( 4 x ) and asked 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 the Outer Function First, we find the derivative of the outer function, which is 4 sec ( 4 x ) . The derivative of sec ( 4 x ) is sec ( 4 x ) tan ( 4 x ) . So, we have 4 ⋅ sec ( 4 x ) tan ( 4 x ) .
Differentiating the Inner Function Next, we need to multiply by the derivative of the inner function, 4 x . The derivative of 4 x with respect to x is 4.
Combining the Results Now, we multiply the result from step 3 by the derivative of the inner function (step 4): f ′ ( x ) = 4 ⋅ sec ( 4 x ) tan ( 4 x ) ⋅ 4
Simplifying the Expression Finally, we simplify the expression: f ′ ( x ) = 16 sec ( 4 x ) tan ( 4 x )
Final Answer Therefore, the derivative of f ( x ) = 4 sec ( 4 x ) is f ′ ( x ) = 16 sec ( 4 x ) tan ( 4 x ) .
Examples
Understanding derivatives of trigonometric functions is crucial in many fields. For example, in physics, if f ( x ) represents the angular position of a rotating object at time x , then f ′ ( x ) gives the angular velocity of the object. If f ( x ) = 4 sec ( 4 x ) , then f ′ ( x ) = 16 sec ( 4 x ) tan ( 4 x ) represents how the angular velocity changes with time, which is important in analyzing rotational motion.
