Derivative of 5a*sin(5*x)*x

1. Apply the product rule:

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

   $f{\left(x \right)} = 5 a \sin{\left(5 x \right)}$; to find $\frac{d}{d x} f{\left(x \right)}$:

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

      1. Let $u = 5 x$.

      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 $\frac{d}{d x} 5 x$:

         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: $5$

         The result of the chain rule is:

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

      So, the result is: $25 a \cos{\left(5 x \right)}$

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

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

   The result is: $25 a x \cos{\left(5 x \right)} + 5 a \sin{\left(5 x \right)}$

2. Now simplify:

   $$5 a \left(5 x \cos{\left(5 x \right)} + \sin{\left(5 x \right)}\right)$$

The answer is:

$$5 a \left(5 x \cos{\left(5 x \right)} + \sin{\left(5 x \right)}\right)$$