Derivative of y=x^3sinx*lnx

1. Apply the product rule:

   $$\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} h{\left(x \right)} = f{\left(x \right)} g{\left(x \right)} \frac{d}{d x} h{\left(x \right)} + f{\left(x \right)} h{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} h{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$$

   $f{\left(x \right)} = x^{3}$; to find $\frac{d}{d x} f{\left(x \right)}$:

   1. Apply the power rule: $x^{3}$ goes to $3 x^{2}$

   $g{\left(x \right)} = \sin{\left(x \right)}$; to find $\frac{d}{d x} g{\left(x \right)}$:

   1. The derivative of sine is cosine:

      $$\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$$

   $h{\left(x \right)} = \log{\left(x \right)}$; to find $\frac{d}{d x} h{\left(x \right)}$:

   1. The derivative of $\log{\left(x \right)}$ is $\frac{1}{x}$.

   The result is: $x^{3} \log{\left(x \right)} \cos{\left(x \right)} + 3 x^{2} \log{\left(x \right)} \sin{\left(x \right)} + x^{2} \sin{\left(x \right)}$

2. Now simplify:

   $$x^{2} \left(x \log{\left(x \right)} \cos{\left(x \right)} + 3 \log{\left(x \right)} \sin{\left(x \right)} + \sin{\left(x \right)}\right)$$

The answer is: