Derivative of cosx-log5x

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

   1. The derivative of cosine is negative sine:

      $$\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}$$

   2. 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: $- \frac{1}{x}$

   The result is: $- \sin{\left(x \right)} - \frac{1}{x}$

The answer is: