Derivative of f(x)=5x²(x+47)

1. Apply the product rule:

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

   $f{\left(x \right)} = 5 x^{2}$; 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. Apply the power rule: $x^{2}$ goes to $2 x$

      So, the result is: $10 x$

   $g{\left(x \right)} = x + 47$; to find $\frac{d}{d x} g{\left(x \right)}$:

   1. Differentiate $x + 47$ term by term:

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

      2. The derivative of the constant $47$ is zero.

      The result is: $1$

   The result is: $5 x^{2} + 10 x \left(x + 47\right)$

2. Now simplify:

   $$5 x \left(3 x + 94\right)$$

The answer is:

$$5 x \left(3 x + 94\right)$$