Find the derivative of [tex]$f(x)=x^3 \cos x$[/tex]

Apply the product rule: f ′ ( x ) = u ′ ( x ) v ( x ) + u ( x ) v ′ ( x ) , where u ( x ) = x 3 and v ( x ) = cos x .
Find the derivatives: u ′ ( x ) = 3 x 2 and v ′ ( x ) = − sin x .
Substitute into the product rule: f ′ ( x ) = ( 3 x 2 ) ( cos x ) + ( x 3 ) ( − sin x ) .
Simplify to get the final answer: f ′ ( x ) = 3 x 2 cos x − x 3 sin x ​ .

Explanation

Problem Setup We are given the function f ( x ) = x 3 cos x and we want to find its derivative.

Applying the Product Rule To find the derivative of f ( x ) , we will use the product rule. The product rule states that if we have a function f ( x ) = u ( x ) v ( x ) , then the derivative f ′ ( x ) is given by f ′ ( x ) = u ′ ( x ) v ( x ) + u ( x ) v ′ ( x ) . In our case, let u ( x ) = x 3 and v ( x ) = cos x .

Derivative of u(x) First, we find the derivative of u ( x ) = x 3 . Using the power rule, we have u ′ ( x ) = 3 x 2 .

Derivative of v(x) Next, we find the derivative of v ( x ) = cos x . The derivative of cos x is − sin x , so v ′ ( x ) = − sin x .

Substituting into the Product Rule Now, we substitute u ( x ) , v ( x ) , u ′ ( x ) , and v ′ ( x ) into the product rule formula:


f ′ ( x ) = u ′ ( x ) v ( x ) + u ( x ) v ′ ( x ) = ( 3 x 2 ) ( cos x ) + ( x 3 ) ( − sin x )

Simplifying the Expression Finally, we simplify the expression:

f ′ ( x ) = 3 x 2 cos x − x 3 sin x
So, the derivative of f ( x ) = x 3 cos x is f ′ ( x ) = 3 x 2 cos x − x 3 sin x .

Final Answer Therefore, the derivative of the function f ( x ) = x 3 cos x is:

f ′ ( x ) = 3 x 2 cos x − x 3 sin x
Examples
Understanding derivatives is crucial in physics, especially when analyzing motion. For example, if x ( t ) = t 3 cos 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 . In this case, v ( t ) = 3 t 2 cos t − t 3 sin t . This allows us to determine the object's instantaneous velocity at any given time, which is essential for predicting its future position and behavior.

Answered by GinnyAnswer | 2025-07-08

The derivative of the function f ( x ) = x 3 cos x is found using the product rule, resulting in f ′ ( x ) = 3 x 2 cos x − x 3 sin x .
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Answered by Anonymous | 2025-07-09