Derivative of y=e^x-6x^7+5log5x+12

1. Differentiate $- 6 x^{7} + e^{x} + 5 \log{\left(5 x \right)} + 12$ term by term:

   1. The derivative of $e^{x}$ is itself.

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

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

         1. Apply the power rule: $x^{7}$ goes to $7 x^{6}$

         So, the result is: $42 x^{6}$

      So, the result is: $- 42 x^{6}$

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

      1. Let $u = 5 x$.

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

      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:

         $$\frac{1}{x}$$

      So, the result is: $\frac{5}{x}$

   4. The derivative of the constant $12$ is zero.

   The result is: $- 42 x^{6} + e^{x} + \frac{5}{x}$

The answer is:

$$- 42 x^{6} + e^{x} + \frac{5}{x}$$