Let $f(x)=x^{6 x}$. Use logarithmic differentiation to determine the derivative.
$f^{\prime}(x)=$
$\square$
$f^{\prime}(1)=$
$\square$
Answer (2)
Take the natural logarithm: ln ( f ( x )) = 6 x ln ( x ) .
Differentiate both sides: f ( x ) f ′ ( x ) = 6 ln ( x ) + 6 .
Find the derivative: f ′ ( x ) = x 6 x ( 6 ln ( x ) + 6 ) .
Evaluate at x = 1 : f ′ ( 1 ) = 6 .
Explanation
Problem Analysis We are given the function f ( x ) = x 6 x and asked to find its derivative f ′ ( x ) using logarithmic differentiation, and then evaluate f ′ ( 1 ) .
Taking Natural Logarithm First, take the natural logarithm of both sides of the equation: ln ( f ( x )) = ln ( x 6 x )
Simplifying the Logarithm Using the power rule of logarithms, we simplify the right side: ln ( f ( x )) = 6 x ln ( x )
Differentiating Both Sides Now, differentiate both sides with respect to x . We use the chain rule on the left side and the product rule on the right side: f ( x ) f ′ ( x ) = 6 ln ( x ) + 6 x ⋅ x 1
Simplifying the Derivative Simplify the right side: f ( x ) f ′ ( x ) = 6 ln ( x ) + 6
Isolating f'(x) Multiply both sides by f ( x ) to isolate f ′ ( x ) :
f ′ ( x ) = f ( x ) ( 6 ln ( x ) + 6 )
Finding the Derivative Substitute f ( x ) = x 6 x into the equation: f ′ ( x ) = x 6 x ( 6 ln ( x ) + 6 ) Thus, the derivative is f ′ ( x ) = x 6 x ( 6 ln ( x ) + 6 ) .
Evaluating f'(1) Now, evaluate f ′ ( 1 ) by substituting x = 1 into the expression for f ′ ( x ) :
f ′ ( 1 ) = 1 6 ( 1 ) ( 6 ln ( 1 ) + 6 ) Since ln ( 1 ) = 0 , we have: f ′ ( 1 ) = 1 ( 6 ( 0 ) + 6 ) = 6 Thus, f ′ ( 1 ) = 6 .
Final Answer Therefore, the derivative of f ( x ) = x 6 x is f ′ ( x ) = x 6 x ( 6 ln ( x ) + 6 ) , and f ′ ( 1 ) = 6 .
Examples
Logarithmic differentiation is particularly useful in analyzing growth rates in various fields. For instance, in finance, it can help model the growth of investments where the growth rate depends on the current investment level. If we consider an investment growing according to V ( t ) = t 6 t , where V ( t ) is the value of the investment at time t , finding V ′ ( t ) using logarithmic differentiation helps in understanding how the investment's growth rate changes over time. This method simplifies complex derivative calculations and provides insights into dynamic systems.
Using logarithmic differentiation, we find that the derivative of f ( x ) = x 6 x is f ′ ( x ) = x 6 x ( 6 ln ( x ) + 6 ) , and when evaluated at x = 1 , it gives f ′ ( 1 ) = 6 .
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