Differentiate the function.
[tex]
\begin{array}{r}
f(x)=\sin (7 \ln (x)) \\
f^{\prime}(x)=\square
\end{array}
[/tex]
Answer (1)
Apply the chain rule to differentiate f ( x ) = sin ( 7 ln ( x )) .
Find the derivative of the outer function: d u d sin ( u ) = cos ( u ) , where u = 7 ln ( x ) .
Find the derivative of the inner function: d x d ( 7 ln ( x )) = x 7 .
Multiply the derivatives: f ′ ( x ) = x 7 c o s ( 7 l n ( x )) .
f ′ ( x ) = x 7 cos ( 7 ln ( x ))
Explanation
Problem Analysis We are given the function f ( x ) = sin ( 7 ln ( x )) and we need to find its derivative f ′ ( x ) . This requires the use of the chain rule.
Applying the Chain Rule The chain rule states that if we have a composite function f ( g ( x )) , then its derivative is f ′ ( g ( x )) ⋅ g ′ ( x ) . In our case, we can consider f ( u ) = sin ( u ) and g ( x ) = 7 ln ( x ) .
Derivative of the Outer Function First, we find the derivative of the outer function f ( u ) = sin ( u ) with respect to u , which is f ′ ( u ) = cos ( u ) . So, f ′ ( g ( x )) = cos ( 7 ln ( x )) .
Derivative of the Inner Function Next, we find the derivative of the inner function g ( x ) = 7 ln ( x ) with respect to x . The derivative of ln ( x ) is x 1 , so the derivative of 7 ln ( x ) is g ′ ( x ) = 7 ⋅ x 1 = x 7 .
Combining the Derivatives Now, we multiply the derivatives of the outer and inner functions: f ′ ( x ) = f ′ ( g ( x )) ⋅ g ′ ( x ) = cos ( 7 ln ( x )) ⋅ x 7 .
Final Answer Finally, we simplify the expression to get f ′ ( x ) = x 7 c o s ( 7 l n ( x )) .
Examples
In physics, you might encounter this type of function when modeling damped oscillations where the frequency changes logarithmically over time. For example, the angular displacement of a pendulum with decreasing amplitude could be modeled using a function similar to the one in this problem. Finding the derivative helps to understand the rate of change of the displacement at any given time, which is crucial for analyzing the system's dynamics.
