If y = x Sin x, find $\frac{dy}{dx}$ when x = $\frac{\pi}{2}$

Analysis

To solve this problem, we need to find the derivative of the function $y = x \sin x$ with respect to $x$. Since $y$ is the product of two functions of $x$ ($x$ and $\sin x$), we must use the Product Rule of differentiation.

Concept: The Product Rule

The product rule states that if $y = u \cdot v$, where $u$ and $v$ are functions of $x$, then:

In this case:

• $u = x$
• $v = \sin x$

Step-by-Step Solution

Step 1: Differentiate the individual parts.

• $\frac{du}{dx} = \frac{d}{dx}(x) = 1$
• $\frac{dv}{dx} = \frac{d}{dx}(\sin x) = \cos x$

Step 2: Apply the Product Rule formula.

Step 3: Substitute $x = \frac{\pi}{2}$ into the derivative.

Step 4: Evaluate the trigonometric functions.

• Recall that $\cos(\frac{\pi}{2}) = 0$
• Recall that $\sin(\frac{\pi}{2}) = 1$

Substituting these values:

Conclusion

The value of the derivative at $x = \frac{\pi}{2}$ is 1. This matches option C.

Exam Tip

When dealing with trigonometric functions in calculus, always ensure your calculator is in Radian mode if you are calculating values, though knowing your unit circle values ($\sin 90^\circ=1, \cos 90^\circ=0$) is much faster and prevents errors during JAMB exams!