Derivative of y=sin(5x)*((2x^5)+53x-5)

1. Apply the product rule:

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

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

   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:

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

   $g{\left(x \right)} = 2 x^{5} + 53 x - 5$; to find $\frac{d}{d x} g{\left(x \right)}$:

   1. Differentiate $2 x^{5} + 53 x - 5$ 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: $x^{5}$ goes to $5 x^{4}$

         So, the result is: $10 x^{4}$

      2. 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: $53$

      3. The derivative of the constant $\left(-1\right) 5$ is zero.

      The result is: $10 x^{4} + 53$

   The result is: $\left(10 x^{4} + 53\right) \sin{\left(5 x \right)} + 5 \cdot \left(2 x^{5} + 53 x - 5\right) \cos{\left(5 x \right)}$

2. Now simplify:

   $$\left(10 x^{4} + 53\right) \sin{\left(5 x \right)} + \left(10 x^{5} + 265 x - 25\right) \cos{\left(5 x \right)}$$

The answer is:

$$\left(10 x^{4} + 53\right) \sin{\left(5 x \right)} + \left(10 x^{5} + 265 x - 25\right) \cos{\left(5 x \right)}$$