Step 1: Understanding the Concept:
When a function is expressed as the product of two simpler functions, we cannot differentiate them individually and multiply the results.
Instead, we must use the Product Rule (Leibniz's Rule).
The rule states that the derivative of a product \(u(x) \cdot v(x)\) is the derivative of the first function times the second function, plus the first function times the derivative of the second function.
Step 2: Key Formula or Approach:
Formula: \(\frac{d}{dx}(uv) = u \frac{dv}{dx} + v \frac{du}{dx}\)
In our case, the given function is \(y = x^2 \sin x\).
We can identify:
\(u = x^2\)
\(v = \sin x\)
Step 3: Detailed Explanation:
Let's find the derivatives of the individual parts first:
Step 3.1: Differentiate \(u\):
Using the power rule \(\frac{d}{dx}(x^n) = nx^{n-1}\):
\[ \frac{du}{dx} = \frac{d}{dx}(x^2) = 2x \]
Step 3.2: Differentiate \(v\):
Using standard trigonometric derivative formulas:
\[ \frac{dv}{dx} = \frac{d}{dx}(\sin x) = \cos x \]
Step 3.3: Apply the Product Rule:
Substitute the components into the formula \(\frac{dy}{dx} = u \frac{dv}{dx} + v \frac{du}{dx}\):
\[ \frac{dy}{dx} = (x^2)(\cos x) + (\sin x)(2x) \]
\[ \frac{dy}{dx} = x^2 \cos x + 2x \sin x \]
Step 3.4: Final Arrangement:
To match the standard formatting in the options, we can reorder the terms:
\[ \frac{dy}{dx} = 2x \sin x + x^2 \cos x \]
Step 4: Final Answer:
The derivative of the function \(y = x^2 \sin x\) is \(2x \sin x + x^2 \cos x\).