Question

product rule
related content
$\frac{d}{dx}(x^{3}sin(x))=$

Explanation:

Step1: Recall product - rule

The product - rule states that if \(y = u\cdot v\), then \(y^\prime=\frac{d}{dx}(u\cdot v)=u^\prime v + uv^\prime\). Let \(u = x^{3}\) and \(v=\sin(x)\).

Step2: Differentiate \(u\) and \(v\)

Differentiate \(u = x^{3}\) with respect to \(x\): \(\frac{d}{dx}(x^{3})=3x^{2}\) (using the power - rule \(\frac{d}{dx}(x^{n})=nx^{n - 1}\)). Differentiate \(v=\sin(x)\) with respect to \(x\): \(\frac{d}{dx}(\sin(x))=\cos(x)\).

Step3: Apply product - rule

Substitute \(u\), \(u^\prime\), \(v\), and \(v^\prime\) into the product - rule formula. \(\frac{d}{dx}(x^{3}\sin(x))=\frac{d}{dx}(x^{3})\sin(x)+x^{3}\frac{d}{dx}(\sin(x))=3x^{2}\sin(x)+x^{3}\cos(x)\).

Answer:

\(3x^{2}\sin(x)+x^{3}\cos(x)\)