Derivative of y=log5(x)-√xx^6

1. Differentiate $- \left(\sqrt{x x}\right)^{6} + \frac{\log{\left(x \right)}}{\log{\left(5 \right)}}$ term by term:

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

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

      So, the result is: $\frac{1}{x \log{\left(5 \right)}}$

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

      1. Let $u = \sqrt{x x}$.

      2. Apply the power rule: $u^{6}$ goes to $6 u^{5}$

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

         1. Let $u = x x$.

         2. Apply the power rule: $\sqrt{u}$ goes to $\frac{1}{2 \sqrt{u}}$

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

            1. Apply the product rule:

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

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

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

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

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

               The result is: $2 x$

            The result of the chain rule is:

            $$\frac{x}{\left|{x}\right|}$$

         The result of the chain rule is:

         $$6 x^{5}$$

      So, the result is: $- 6 x^{5}$

   The result is: $- 6 x^{5} + \frac{1}{x \log{\left(5 \right)}}$

The answer is:

$$- 6 x^{5} + \frac{1}{x \log{\left(5 \right)}}$$