Differentiate with respect to [tex]$x$[/tex]
[tex]$\frac{e^{2 x^2} \sqrt{\sin x}}{(2 x+1)^3}=$[/tex]
Answer (2)
Take the natural logarithm of the function: ln ( f ( x )) = 2 x 2 + 2 1 ln ( sin x ) − 3 ln ( 2 x + 1 ) .
Differentiate both sides with respect to x : f ( x ) f ′ ( x ) = 4 x + 2 1 cot x − 2 x + 1 6 .
Multiply by f ( x ) to find the derivative: f ′ ( x ) = f ( x ) ( 4 x + 2 1 cot x − 2 x + 1 6 ) .
Substitute f ( x ) to get the final answer: f ′ ( x ) = ( 2 x + 1 ) 3 e 2 x 2 s i n x ( 4 x + 2 1 cot x − 2 x + 1 6 ) .
Explanation
Problem Analysis We are asked to find the derivative of the function f ( x ) = ( 2 x + 1 ) 3 e 2 x 2 s i n x with respect to x . This looks like a job for logarithmic differentiation!
Taking the Natural Logarithm First, take the natural logarithm of both sides of the equation f ( x ) = ( 2 x + 1 ) 3 e 2 x 2 s i n x . This gives us
ln ( f ( x )) = ln ( ( 2 x + 1 ) 3 e 2 x 2 sin x )
Using properties of logarithms, we can expand the right side:
ln ( f ( x )) = ln ( e 2 x 2 ) + ln ( sin x ) − ln (( 2 x + 1 ) 3 )
ln ( f ( x )) = 2 x 2 + 2 1 ln ( sin x ) − 3 ln ( 2 x + 1 )
Differentiating Both Sides Now, differentiate both sides of the equation with respect to x . Remember that d x d ln ( f ( x )) = f ( x ) f ′ ( x ) . So we have:
f ( x ) f ′ ( x ) = d x d ( 2 x 2 + 2 1 ln ( sin x ) − 3 ln ( 2 x + 1 ) )
f ( x ) f ′ ( x ) = 4 x + 2 1 ⋅ sin x cos x − 3 ⋅ 2 x + 1 2
f ( x ) f ′ ( x ) = 4 x + 2 1 cot x − 2 x + 1 6
Isolating f'(x) Next, multiply both sides by f ( x ) to isolate f ′ ( x ) :
f ′ ( x ) = f ( x ) ( 4 x + 2 1 cot x − 2 x + 1 6 )
Substitute the original function f ( x ) = ( 2 x + 1 ) 3 e 2 x 2 s i n x back into the equation:
f ′ ( x ) = ( 2 x + 1 ) 3 e 2 x 2 sin x ( 4 x + 2 1 cot x − 2 x + 1 6 )
Final Answer Therefore, the derivative of the given function is:
f ′ ( x ) = ( 2 x + 1 ) 3 e 2 x 2 sin x ( 4 x + 2 1 cot x − 2 x + 1 6 )
Examples
Logarithmic differentiation is particularly useful in analyzing growth rates in various fields. For instance, in economics, it can be used to determine the elasticity of demand, which measures how much the quantity demanded of a good responds to a change in its price. Similarly, in biology, it can help model population growth rates that depend on multiple factors, such as birth rates, death rates, and resource availability. By taking logarithms and differentiating, complex multiplicative relationships can be transformed into simpler additive ones, making analysis and interpretation easier.
To differentiate the function f ( x ) = ( 2 x + 1 ) 3 e 2 x 2 s i n x , we use logarithmic differentiation. By taking the natural logarithm and differentiating, we isolate f ′ ( x ) to get: f ′ ( x ) = ( 2 x + 1 ) 3 e 2 x 2 s i n x ( 4 x + 2 1 cot x − 2 x + 1 6 ) .
;
