Differentiate the function. [tex]y=\log _4\left(x \log _7(x)\right)[/tex] [tex]y^{\prime}=[/tex]
Answer (2)
Rewrite the function using the change of base formula: y = l n ( 4 ) l n ( x l o g 7 ( x )) .
Apply the chain rule and product rule to differentiate the function.
Simplify the derivative to obtain y ′ = x l n ( 4 ) l n ( x ) 1 + l n ( x ) .
The final answer is: x ln ( 4 ) ln ( x ) 1 + ln ( x ) .
Explanation
Problem Analysis We are given the function y = lo g 4 ( x lo g 7 ( x )) and asked to find its derivative y ′ . To do this, we will use the chain rule and properties of logarithms.
Rewriting the Function First, let's rewrite the function using the change of base formula for logarithms. We can change the base to the natural logarithm (ln) as follows: y = lo g 4 ( x lo g 7 ( x )) = ln ( 4 ) ln ( x lo g 7 ( x )) Now, we can further rewrite lo g 7 ( x ) as l n ( 7 ) l n ( x ) , so y = ln ( 4 ) ln ( x l n ( 7 ) l n ( x ) ) = ln ( 4 ) ln ( x ) + ln ( l n ( 7 ) l n ( x ) ) = ln ( 4 ) ln ( x ) + ln ( ln ( x )) − ln ( ln ( 7 ))
Differentiating the Function Now, we differentiate y with respect to x . Using the chain rule and the fact that the derivative of ln ( x ) is x 1 , we have: d x d y = ln ( 4 ) 1 [ x 1 + ln ( x ) 1 ⋅ x 1 − 0 ] = ln ( 4 ) 1 [ x 1 + x ln ( x ) 1 ] Combining the terms inside the brackets, we get: d x d y = ln ( 4 ) 1 [ x ln ( x ) ln ( x ) + 1 ] = x ln ( 4 ) ln ( x ) 1 + ln ( x )
Final Answer Thus, the derivative of the function is: y ′ = x ln ( 4 ) ln ( x ) 1 + ln ( x )
Examples
Logarithmic differentiation is incredibly useful in fields like finance, where understanding rates of change in investments or loans is crucial. For instance, if you're analyzing an investment portfolio whose growth depends on multiple logarithmic factors, being able to differentiate such a function allows you to precisely determine how sensitive the portfolio's growth is to changes in each factor. This helps in making informed decisions about risk management and investment strategies.
To differentiate the function y = lo g 4 ( x lo g 7 ( x )) , we rewrite it in terms of natural logarithms and use the chain and product rules. The final derivative is given by y ′ = x l n ( 4 ) l n ( x ) 1 + l n ( x ) . This process illustrates how to apply logarithmic properties along with differentiation rules effectively.
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