Derivative of x^5lnx+5x

1. Differentiate $x^{5} \log{\left(x \right)} + 5 x$ term by term:

   1. Apply the product rule:

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

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

      1. Apply the power rule: $x^{5}$ goes to $5 x^{4}$

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

      1. The derivative of $\log{\left(x \right)}$ is $\frac{1}{x}$.

      The result is: $5 x^{4} \log{\left(x \right)} + x^{4}$

   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: $5$

   The result is: $5 x^{4} \log{\left(x \right)} + x^{4} + 5$

The answer is:

$$5 x^{4} \log{\left(x \right)} + x^{4} + 5$$