Derivative of y=x^(2)sin(5x-3)

1. Apply the product rule:

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

   $f{\left(x \right)} = x^{2}$; to find $\frac{d}{d x} f{\left(x \right)}$:

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

   $g{\left(x \right)} = \sin{\left(5 x - 3 \right)}$; to find $\frac{d}{d x} g{\left(x \right)}$:

   1. Let $u = 5 x - 3$.

   2. The derivative of sine is cosine:

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

   3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(5 x - 3\right)$:

      1. Differentiate $5 x - 3$ term by term:

         1. The derivative of a constant times a function is the constant times the derivative of the function.

            1. Apply the power rule: $x$ goes to $1$

            So, the result is: $5$

         2. The derivative of the constant $-3$ is zero.

         The result is: $5$

      The result of the chain rule is:

      $$5 \cos{\left(5 x - 3 \right)}$$

   The result is: $5 x^{2} \cos{\left(5 x - 3 \right)} + 2 x \sin{\left(5 x - 3 \right)}$

2. Now simplify:

   $$x \left(5 x \cos{\left(5 x - 3 \right)} + 2 \sin{\left(5 x - 3 \right)}\right)$$

The answer is:

$$x \left(5 x \cos{\left(5 x - 3 \right)} + 2 \sin{\left(5 x - 3 \right)}\right)$$