Derivative of y*e^(5y)

1. Apply the product rule:

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

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

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

   $g{\left(y \right)} = e^{5 y}$; to find $\frac{d}{d y} g{\left(y \right)}$:

   1. Let $u = 5 y$.

   2. The derivative of $e^{u}$ is itself.

   3. Then, apply the chain rule. Multiply by $\frac{d}{d y} 5 y$:

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

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

         So, the result is: $5$

      The result of the chain rule is:

      $$5 e^{5 y}$$

   The result is: $5 y e^{5 y} + e^{5 y}$

2. Now simplify:

   $$\left(5 y + 1\right) e^{5 y}$$

The answer is:

$$\left(5 y + 1\right) e^{5 y}$$