Derivative of y=logx-cosecx/5+5-3/x√x

1. Differentiate $- \frac{3}{x} \sqrt{x} + \left(\left(- \frac{x c \cos{\left(e \right)}}{5} + \log{\left(x \right)}\right) + 5\right)$ term by term:

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

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

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

         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: $x$ goes to $1$

               So, the result is: $c \cos{\left(e \right)}$

            So, the result is: $- \frac{c \cos{\left(e \right)}}{5}$

         The result is: $- \frac{c \cos{\left(e \right)}}{5} + \frac{1}{x}$

      2. The derivative of the constant $5$ is zero.

      The result is: $- \frac{c \cos{\left(e \right)}}{5} + \frac{1}{x}$

   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 quotient rule, which is:

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

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

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

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

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

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

            Now plug in to the quotient rule:

            $- \frac{1}{2 x^{\frac{3}{2}}}$

         So, the result is: $- \frac{3}{2 x^{\frac{3}{2}}}$

      So, the result is: $\frac{3}{2 x^{\frac{3}{2}}}$

   The result is: $- \frac{c \cos{\left(e \right)}}{5} + \frac{1}{x} + \frac{3}{2 x^{\frac{3}{2}}}$

The answer is:

$$- \frac{c \cos{\left(e \right)}}{5} + \frac{1}{x} + \frac{3}{2 x^{\frac{3}{2}}}$$