Derivative of y=ctg(x)+sqrt(x)

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           ___
cot(x) + \/ x

\sqrt{x} + \cot{\left(x \right)}

d /           ___\
--\cot(x) + \/ x /
dx

\frac{d}{d x} \left(\sqrt{x} + \cot{\left(x \right)}\right)

Transform

Detail solution

Differentiate $\sqrt{x} + \cot{\left(x \right)}$ term by term:
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 $\frac{d}{d x} f{\left(x \right)}$ :
      
      The derivative of sine is cosine:
      
      $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
      
      To find $\frac{d}{d x} g{\left(x \right)}$ :
      
      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:
    
    $\frac{- \sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}$
2. Apply the power rule: $\sqrt{x}$ goes to $\frac{1}{2 \sqrt{x}}$
The result is: $- \frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)} \tan^{2}{\left(x \right)}} + \frac{1}{2 \sqrt{x}}$
Now simplify:

$- \frac{1}{\sin^{2}{\left(x \right)}} + \frac{1}{2 \sqrt{x}}$

The answer is:

- \frac{1}{\sin^{2}{\left(x \right)}} + \frac{1}{2 \sqrt{x}}

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

        1         2   
-1 + ------- - cot (x)
         ___          
     2*\/ x

- \cot^{2}{\left(x \right)} - 1 + \frac{1}{2 \sqrt{x}}

Simplify

The second derivative [src]

    1        /       2   \       
- ------ + 2*\1 + cot (x)/*cot(x)
     3/2                         
  4*x

2 \left(\cot^{2}{\left(x \right)} + 1\right) \cot{\left(x \right)} - \frac{1}{4 x^{\frac{3}{2}}}

Simplify

The third derivative [src]

                 2                                   
    /       2   \      3           2    /       2   \
- 2*\1 + cot (x)/  + ------ - 4*cot (x)*\1 + cot (x)/
                        5/2                          
                     8*x

- 2 \left(\cot^{2}{\left(x \right)} + 1\right)^{2} - 4 \left(\cot^{2}{\left(x \right)} + 1\right) \cot^{2}{\left(x \right)} + \frac{3}{8 x^{\frac{5}{2}}}

Simplify

The graph

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Derivative of y=ctg(x)+sqrt(x)

The graph:

Piecewise:

The solution

Method #1

Method #2