Differentiate the function. [tex]F(s)=\ln (\ln (4 s))[/tex] [tex]F^{\prime}(s)=[/tex]
Answer (1)
Apply the chain rule by setting u = ln ( 4 s ) , transforming the function to F ( s ) = ln ( u ) .
Find the derivative of u with respect to s , which is u ′ ( s ) = s 1 .
Substitute u and u ′ ( s ) back into the chain rule formula, resulting in F ′ ( s ) = l n ( 4 s ) 1 ⋅ s 1 .
Simplify the expression to obtain the final derivative: F ′ ( s ) = s ln ( 4 s ) 1 .
Explanation
Problem Analysis We are asked to find the derivative of the function F ( s ) = ln ( ln ( 4 s )) . This requires applying the chain rule.
Applying the Chain Rule Let u = ln ( 4 s ) . Then F ( s ) = ln ( u ) . By the chain rule, F ′ ( s ) = u 1 ⋅ u ′ ( s ) .
Finding the Derivative of u(s) Now we need to find u ′ ( s ) . Since u = ln ( 4 s ) , we have u ′ ( s ) = 4 s 1 ⋅ 4 = s 1 .
Substituting Back Substituting u = ln ( 4 s ) and u ′ ( s ) = s 1 into F ′ ( s ) = u 1 ⋅ u ′ ( s ) , we get F ′ ( s ) = l n ( 4 s ) 1 ⋅ s 1 = s l n ( 4 s ) 1 .
Final Answer Therefore, the derivative of F ( s ) = ln ( ln ( 4 s )) is F ′ ( s ) = s l n ( 4 s ) 1 .
Examples
In population modeling, if s represents time and F ( s ) represents the growth rate of a population, then F ′ ( s ) gives the rate of change of the growth rate. Understanding how to differentiate such logarithmic functions helps in predicting future population trends and managing resources effectively.
