Use logarithmic differentiation to find the derivative of the function.
$y=\left(x^5+2\right)^2\left(x^4+4\right)^4$
$y^{\prime}(x)=$
$\square$
Answer (1)
Take the natural logarithm of both sides: ln y = 2 ln ( x 5 + 2 ) + 4 ln ( x 4 + 4 ) .
Differentiate both sides with respect to x : y 1 d x d y = x 5 + 2 10 x 4 + x 4 + 4 16 x 3 .
Multiply both sides by y : d x d y = ( x 5 + 2 ) 2 ( x 4 + 4 ) 4 ( x 5 + 2 10 x 4 + x 4 + 4 16 x 3 ) .
Simplify the expression: y ′ ( x ) = ( x 5 + 2 ) ( x 4 + 4 ) 3 ( 26 x 8 + 32 x 3 + 40 x 4 ) .
Explanation
Problem Analysis and Strategy We are given the function y = ( x 5 + 2 ) 2 ( x 4 + 4 ) 4 and we want to find its derivative y ′ using logarithmic differentiation. Logarithmic differentiation is a technique used to differentiate functions that involve products, quotients, or exponents, especially when these operations are intertwined in a complex way. It simplifies the differentiation process by converting multiplication and division into addition and subtraction through the use of logarithms.
Taking the Natural Logarithm First, take the natural logarithm of both sides of the equation: ln y = ln [ ( x 5 + 2 ) 2 ( x 4 + 4 ) 4 ] Using logarithm properties, we can expand the right side: ln y = 2 ln ( x 5 + 2 ) + 4 ln ( x 4 + 4 )
Differentiating Both Sides Now, differentiate both sides with respect to x . Remember to use the chain rule: y 1 d x d y = 2 ⋅ x 5 + 2 1 ⋅ 5 x 4 + 4 ⋅ x 4 + 4 1 ⋅ 4 x 3 Simplifying the right side: y 1 d x d y = x 5 + 2 10 x 4 + x 4 + 4 16 x 3
Isolating dy/dx Multiply both sides by y to isolate d x d y :
d x d y = y ( x 5 + 2 10 x 4 + x 4 + 4 16 x 3 ) Substitute the original expression for y :
d x d y = ( x 5 + 2 ) 2 ( x 4 + 4 ) 4 ( x 5 + 2 10 x 4 + x 4 + 4 16 x 3 )
Simplifying the Expression To simplify the expression, find a common denominator for the terms inside the parentheses: d x d y = ( x 5 + 2 ) 2 ( x 4 + 4 ) 4 ( ( x 5 + 2 ) ( x 4 + 4 ) 10 x 4 ( x 4 + 4 ) + 16 x 3 ( x 5 + 2 ) ) d x d y = ( x 5 + 2 ) 2 ( x 4 + 4 ) 4 ( ( x 5 + 2 ) ( x 4 + 4 ) 10 x 8 + 40 x 4 + 16 x 8 + 32 x 3 ) d x d y = ( x 5 + 2 ) 2 ( x 4 + 4 ) 4 ( ( x 5 + 2 ) ( x 4 + 4 ) 26 x 8 + 32 x 3 + 40 x 4 )
Final Simplification Now, simplify by canceling out one factor of ( x 5 + 2 ) and one factor of ( x 4 + 4 ) :
d x d y = ( x 5 + 2 ) ( x 4 + 4 ) 3 ( 26 x 8 + 32 x 3 + 40 x 4 ) So, the final answer is: y ′ ( x ) = ( x 5 + 2 ) ( x 4 + 4 ) 3 ( 26 x 8 + 32 x 3 + 40 x 4 )
Final Answer Therefore, the derivative of the function y = ( x 5 + 2 ) 2 ( x 4 + 4 ) 4 using logarithmic differentiation is: y ′ ( x ) = ( x 5 + 2 ) ( x 4 + 4 ) 3 ( 26 x 8 + 32 x 3 + 40 x 4 )
Examples
Logarithmic differentiation is particularly useful in economics and finance when dealing with functions that model growth rates or compounded effects. For instance, consider a scenario where a company's revenue is modeled by a function involving products of several factors, each dependent on time. Using logarithmic differentiation, economists can easily analyze the proportional rate of change of revenue with respect to time, providing insights into the factors driving revenue growth and making informed decisions about investment and strategy. This method simplifies complex calculations and offers a clear understanding of the relationships between different economic variables.
