If $f(x)=2 x \sin x \cos x$, find
$f^{\prime}(x)=$
Answer (1)
Simplify the function using the identity 2 sin x cos x = sin ( 2 x ) , so f ( x ) = x sin ( 2 x ) .
Apply the product rule: f ′ ( x ) = ( x ) ′ sin ( 2 x ) + x ( sin ( 2 x ) ) ′ .
Find the derivatives: ( x ) ′ = 1 and ( sin ( 2 x ) ) ′ = 2 cos ( 2 x ) .
Substitute and simplify: f ′ ( x ) = sin ( 2 x ) + 2 x cos ( 2 x ) .
f ′ ( x ) = sin ( 2 x ) + 2 x cos ( 2 x )
Explanation
Problem Analysis We are given the function f ( x ) = 2 x sin x cos x and we want to find its derivative f ′ ( x ) .
Simplifying the Function First, we can simplify the function using the trigonometric identity 2 sin x cos x = sin ( 2 x ) . Therefore, we can rewrite the function as f ( x ) = x sin ( 2 x ) .
Applying the Product Rule Now, we need to find the derivative of f ( x ) = x sin ( 2 x ) . We will use the product rule, which states that if f ( x ) = u ( x ) v ( x ) , then f ′ ( x ) = u ′ ( x ) v ( x ) + u ( x ) v ′ ( x ) . In our case, u ( x ) = x and v ( x ) = sin ( 2 x ) .
Finding the Derivatives We find the derivatives of u ( x ) and v ( x ) . The derivative of u ( x ) = x is u ′ ( x ) = 1 . The derivative of v ( x ) = sin ( 2 x ) is v ′ ( x ) = 2 cos ( 2 x ) using the chain rule.
Substituting into the Product Rule Now, we substitute these derivatives into the product rule formula: f ′ ( x ) = u ′ ( x ) v ( x ) + u ( x ) v ′ ( x ) = 1 ⋅ sin ( 2 x ) + x ⋅ 2 cos ( 2 x ) .
Simplifying the Expression Finally, we simplify the expression to get f ′ ( x ) = sin ( 2 x ) + 2 x cos ( 2 x ) .
Final Answer Therefore, the derivative of the function f ( x ) = 2 x sin x cos x is f ′ ( x ) = sin ( 2 x ) + 2 x cos ( 2 x ) .
Examples
Understanding derivatives is crucial in physics, especially when analyzing motion. For instance, if f ( x ) represents the position of an object at time x , then f ′ ( x ) gives the object's velocity at that time. In our case, if f ( x ) = 2 x sin x cos x described a specific oscillatory motion, finding f ′ ( x ) = sin ( 2 x ) + 2 x cos ( 2 x ) would tell us how the velocity changes with time, essential for predicting the object's behavior.
