Find \(\frac{dy}{dx}\) if \(y=6x^2 \sin x \cos x\)
Answer (1)
Rewrite the function using the identity 2 sin x cos x = sin ( 2 x ) to get y = 3 x 2 sin ( 2 x ) .
Apply the product rule: d x d y = u ′ v + u v ′ , where u = 3 x 2 and v = sin ( 2 x ) .
Find the derivatives: u ′ = 6 x and v ′ = 2 cos ( 2 x ) .
Substitute into the product rule and simplify: d x d y = 6 x ( sin ( 2 x ) + x cos ( 2 x )) .
The derivative is 6 x ( sin ( 2 x ) + x cos ( 2 x )) .
Explanation
Problem Setup We are given the function y = 6 x 2 sin x cos x and asked to find its derivative with respect to x , denoted as d x d y .
Simplifying the Function First, we can simplify the function using the trigonometric identity 2 sin x cos x = sin ( 2 x ) . Thus, we can rewrite the function as y = 3 x 2 ( 2 sin x cos x ) = 3 x 2 sin ( 2 x ) .
Applying the Product Rule Now, we need to find the derivative of y = 3 x 2 sin ( 2 x ) with respect to x . We will use the product rule, which states that if y = u ( x ) v ( x ) , then d x d y = u ′ ( x ) v ( x ) + u ( x ) v ′ ( x ) . In our case, let u ( x ) = 3 x 2 and v ( x ) = sin ( 2 x ) .
Finding the Derivatives Next, we find the derivatives of u ( x ) and v ( x ) with respect to x .
d x d u = d x d ( 3 x 2 ) = 6 x
d x d v = d x d ( sin ( 2 x )) = 2 cos ( 2 x )
Applying the Product Rule Now, we apply the product rule:
d x d y = u ′ ( x ) v ( x ) + u ( x ) v ′ ( x ) = ( 6 x ) ( sin ( 2 x )) + ( 3 x 2 ) ( 2 cos ( 2 x )) = 6 x sin ( 2 x ) + 6 x 2 cos ( 2 x )
Simplifying the Expression Finally, we can factor out 6 x from the expression:
d x d y = 6 x ( sin ( 2 x ) + x cos ( 2 x ))
Final Answer Therefore, the derivative of y = 6 x 2 sin x cos x with respect to x is 6 x ( sin ( 2 x ) + x cos ( 2 x )) .
Examples
Understanding derivatives is crucial in physics, especially when analyzing motion. For example, if x ( t ) represents the position of an object at time t , then its velocity v ( t ) is the derivative of x ( t ) with respect to t , i.e., v ( t ) = d t d x . Similarly, the acceleration a ( t ) is the derivative of the velocity with respect to time, a ( t ) = d t d v . These concepts are fundamental in describing and predicting the motion of objects.
