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tan(1/((x-1)^(1/2)))

The derivative of the function/ tan(1/((x+1)^(1/2)))

Derivative of tan(1/((x+1)^(1/2)))

The solution

You have entered [src]

   /    1    \
tan|---------|
   |  _______|
   \\/ x + 1 /

$$\tan{\left(\frac{1}{\sqrt{x + 1}} \right)}$$

tan(1/(sqrt(x + 1)))

Detail solution

Rewrite the function to be differentiated:

$\tan{\left(\frac{1}{\sqrt{x + 1}} \right)} = \frac{\sin{\left(\frac{1}{\sqrt{x + 1}} \right)}}{\cos{\left(\frac{1}{\sqrt{x + 1}} \right)}}$
Apply the quotient rule, which is:

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

$f{\left(x \right)} = \sin{\left(\frac{1}{\sqrt{x + 1}} \right)}$ and $g{\left(x \right)} = \cos{\left(\frac{1}{\sqrt{x + 1}} \right)}$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Let $u = \frac{1}{\sqrt{x + 1}}$ .
2. The derivative of sine is cosine:
  
  $\frac{d}{d u} \sin{\left(u \right)} = \cos{\left(u \right)}$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \frac{1}{\sqrt{x + 1}}$ :
  1. Let $u = \sqrt{x + 1}$ .
  2. Apply the power rule: $\frac{1}{u}$ goes to $- \frac{1}{u^{2}}$
  3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \sqrt{x + 1}$ :
    1. Let $u = x + 1$ .
    2. Apply the power rule: $\sqrt{u}$ goes to $\frac{1}{2 \sqrt{u}}$
    3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(x + 1\right)$ :
      1. Differentiate $x + 1$ term by term:
        
        Apply the power rule: $x$ goes to $1$
        
        The derivative of the constant $1$ is zero.
        
        The result is: $1$
      The result of the chain rule is:
      
      $\frac{1}{2 \sqrt{x + 1}}$
    The result of the chain rule is:
    
    $- \frac{1}{2 \left(x + 1\right)^{\frac{3}{2}}}$
  The result of the chain rule is:
  
  $- \frac{\cos{\left(\frac{1}{\sqrt{x + 1}} \right)}}{2 \left(x + 1\right)^{\frac{3}{2}}}$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Let $u = \frac{1}{\sqrt{x + 1}}$ .
2. The derivative of cosine is negative sine:
  
  $\frac{d}{d u} \cos{\left(u \right)} = - \sin{\left(u \right)}$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \frac{1}{\sqrt{x + 1}}$ :
  1. Let $u = \sqrt{x + 1}$ .
  2. Apply the power rule: $\frac{1}{u}$ goes to $- \frac{1}{u^{2}}$
  3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \sqrt{x + 1}$ :
    1. Let $u = x + 1$ .
    2. Apply the power rule: $\sqrt{u}$ goes to $\frac{1}{2 \sqrt{u}}$
    3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(x + 1\right)$ :
      1. Differentiate $x + 1$ term by term:
        
        Apply the power rule: $x$ goes to $1$
        
        The derivative of the constant $1$ is zero.
        
        The result is: $1$
      The result of the chain rule is:
      
      $\frac{1}{2 \sqrt{x + 1}}$
    The result of the chain rule is:
    
    $- \frac{1}{2 \left(x + 1\right)^{\frac{3}{2}}}$
  The result of the chain rule is:
  
  $\frac{\sin{\left(\frac{1}{\sqrt{x + 1}} \right)}}{2 \left(x + 1\right)^{\frac{3}{2}}}$
Now plug in to the quotient rule:

$\frac{- \frac{\sin^{2}{\left(\frac{1}{\sqrt{x + 1}} \right)}}{2 \left(x + 1\right)^{\frac{3}{2}}} - \frac{\cos^{2}{\left(\frac{1}{\sqrt{x + 1}} \right)}}{2 \left(x + 1\right)^{\frac{3}{2}}}}{\cos^{2}{\left(\frac{1}{\sqrt{x + 1}} \right)}}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

 /       2/    1    \\ 
-|1 + tan |---------|| 
 |        |  _______|| 
 \        \\/ x + 1 // 
-----------------------
               3/2     
      2*(x + 1)

$$- \frac{\tan^{2}{\left(\frac{1}{\sqrt{x + 1}} \right)} + 1}{2 \left(x + 1\right)^{\frac{3}{2}}}$$

Simplify

The second derivative [src]

                      /                  /    1    \\
                      |             2*tan|---------||
                      |                  |  _______||
/       2/    1    \\ |    3             \\/ 1 + x /|
|1 + tan |---------||*|---------- + ----------------|
|        |  _______|| |       5/2              3    |
\        \\/ 1 + x // \(1 + x)          (1 + x)     /
-----------------------------------------------------
                          4

$$\frac{\left(\frac{2 \tan{\left(\frac{1}{\sqrt{x + 1}} \right)}}{\left(x + 1\right)^{3}} + \frac{3}{\left(x + 1\right)^{\frac{5}{2}}}\right) \left(\tan^{2}{\left(\frac{1}{\sqrt{x + 1}} \right)} + 1\right)}{4}$$

Simplify

The third derivative [src]

                       /               /       2/    1    \\        2/    1    \         /    1    \\ 
                       |             2*|1 + tan |---------||   4*tan |---------|   18*tan|---------|| 
                       |               |        |  _______||         |  _______|         |  _______|| 
 /       2/    1    \\ |    15         \        \\/ 1 + x //         \\/ 1 + x /         \\/ 1 + x /| 
-|1 + tan |---------||*|---------- + ----------------------- + ----------------- + -----------------| 
 |        |  _______|| |       7/2                 9/2                    9/2                  4    | 
 \        \\/ 1 + x // \(1 + x)             (1 + x)                (1 + x)              (1 + x)     / 
------------------------------------------------------------------------------------------------------
                                                  8

$$- \frac{\left(\tan^{2}{\left(\frac{1}{\sqrt{x + 1}} \right)} + 1\right) \left(\frac{18 \tan{\left(\frac{1}{\sqrt{x + 1}} \right)}}{\left(x + 1\right)^{4}} + \frac{15}{\left(x + 1\right)^{\frac{7}{2}}} + \frac{2 \left(\tan^{2}{\left(\frac{1}{\sqrt{x + 1}} \right)} + 1\right)}{\left(x + 1\right)^{\frac{9}{2}}} + \frac{4 \tan^{2}{\left(\frac{1}{\sqrt{x + 1}} \right)}}{\left(x + 1\right)^{\frac{9}{2}}}\right)}{8}$$

Simplify

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Derivative of tan(1/((x+1)^(1/2)))

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