Derivative of 5ctgx+8+9sqrt(x)

1. Differentiate $9 \sqrt{x} + \left(5 \cot{\left(x \right)} + 8\right)$ term by term:

   1. Differentiate $5 \cot{\left(x \right)} + 8$ term by term:

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

         1. There are multiple ways to do this derivative.

            Method #1

            1. Rewrite the function to be differentiated:

               $$\cot{\left(x \right)} = \frac{1}{\tan{\left(x \right)}}$$

            2. Let $u = \tan{\left(x \right)}$.

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

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

               1. Rewrite the function to be differentiated:

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

               2. Apply the quotient rule, which is:

                  $$\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$$

                  $f{\left(x \right)} = \sin{\left(x \right)}$ and $g{\left(x \right)} = \cos{\left(x \right)}$.

                  To find $\frac{d}{d x} f{\left(x \right)}$:

                  1. The derivative of sine is cosine:

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

                  To find $\frac{d}{d x} g{\left(x \right)}$:

                  1. The derivative of cosine is negative sine:

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

                  Now plug in to the quotient rule:

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

               The result of the chain rule is:

               $$- \frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)} \tan^{2}{\left(x \right)}}$$

            Method #2

            1. Rewrite the function to be differentiated:

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

            2. Apply the quotient rule, which is:

               $$\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$$

               $f{\left(x \right)} = \cos{\left(x \right)}$ and $g{\left(x \right)} = \sin{\left(x \right)}$.

               To find $\frac{d}{d x} f{\left(x \right)}$:

               1. The derivative of cosine is negative sine:

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

               To find $\frac{d}{d x} g{\left(x \right)}$:

               1. The derivative of sine is cosine:

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

               Now plug in to the quotient rule:

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

         So, the result is: $- \frac{5 \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right)}{\cos^{2}{\left(x \right)} \tan^{2}{\left(x \right)}}$

      2. The derivative of the constant $8$ is zero.

      The result is: $- \frac{5 \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right)}{\cos^{2}{\left(x \right)} \tan^{2}{\left(x \right)}}$

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

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

      So, the result is: $\frac{9}{2 \sqrt{x}}$

   The result is: $- \frac{5 \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right)}{\cos^{2}{\left(x \right)} \tan^{2}{\left(x \right)}} + \frac{9}{2 \sqrt{x}}$

2. Now simplify:

   $$- \frac{5}{\sin^{2}{\left(x \right)}} + \frac{9}{2 \sqrt{x}}$$

The answer is:

$$- \frac{5}{\sin^{2}{\left(x \right)}} + \frac{9}{2 \sqrt{x}}$$