Derivative of 3*sin²x-lgx+3cos²x

1. Differentiate $\left(- \log{\left(x \right)} + 3 \sin^{2}{\left(x \right)}\right) + 3 \cos^{2}{\left(x \right)}$ term by term:

   1. Differentiate $- \log{\left(x \right)} + 3 \sin^{2}{\left(x \right)}$ term by term:

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

         1. Let $u = \sin{\left(x \right)}$.

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

         3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \sin{\left(x \right)}$:

            1. The derivative of sine is cosine:

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

            The result of the chain rule is:

            $$2 \sin{\left(x \right)} \cos{\left(x \right)}$$

         So, the result is: $6 \sin{\left(x \right)} \cos{\left(x \right)}$

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

      The result is: $6 \sin{\left(x \right)} \cos{\left(x \right)} - \frac{1}{x}$

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

      1. Let $u = \cos{\left(x \right)}$.

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

      3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \cos{\left(x \right)}$:

         1. The derivative of cosine is negative sine:

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

         The result of the chain rule is:

         $$- 2 \sin{\left(x \right)} \cos{\left(x \right)}$$

      So, the result is: $- 6 \sin{\left(x \right)} \cos{\left(x \right)}$

   The result is: $- \frac{1}{x}$

The answer is:

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