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Derivative of:
Derivative of sin(x)*sin(x)
Derivative of f
Derivative of log(2*x-1)
Derivative of 4*x^4
Identical expressions
y=ln(sqrt(x^ two + one)-(two - one)*tgx)
y equally ln( square root of (x squared plus 1) minus (2 minus 1) multiply by tgx)
y equally ln( square root of (x to the power of two plus one) minus (two minus one) multiply by tgx)
y=ln(√(x^2+1)-(2-1)*tgx)
y=ln(sqrt(x2+1)-(2-1)*tgx)
y=lnsqrtx2+1-2-1*tgx
y=ln(sqrt(x²+1)-(2-1)*tgx)
y=ln(sqrt(x to the power of 2+1)-(2-1)*tgx)
y=ln(sqrt(x^2+1)-(2-1)tgx)
y=ln(sqrt(x2+1)-(2-1)tgx)
y=lnsqrtx2+1-2-1tgx
y=lnsqrtx^2+1-2-1tgx
Similar expressions
y=ln(sqrt(x^2+1)+(2-1)*tgx)
y=ln(sqrt(x^2+1)-(2+1)*tgx)
y=ln(sqrt(x^2-1)-(2-1)*tgx)

The derivative of the function/ y=ln(sqrt(x^2+1)-(2-1)*tgx)

Derivative of y=ln(sqrt(x^2+1)-(2-1)*tgx)

The solution

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   /   ________         \
   |  /  2              |
log\\/  x  + 1  - tan(x)/

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

Transform

Detail solution

Let $u = \sqrt{x^{2} + 1} - \tan{\left(x \right)}$ .
The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(\sqrt{x^{2} + 1} - \tan{\left(x \right)}\right)$ :
1. Differentiate $\sqrt{x^{2} + 1} - \tan{\left(x \right)}$ term by term:
  1. Let $u = x^{2} + 1$ .
  2. Apply the power rule: $\sqrt{u}$ goes to $\frac{1}{2 \sqrt{u}}$
  3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(x^{2} + 1\right)$ :
    1. Differentiate $x^{2} + 1$ term by term:
      1. Apply the power rule: $x^{2}$ goes to $2 x$
      2. The derivative of the constant $1$ is zero.
      The result is: $2 x$
    The result of the chain rule is:
    
    $\frac{x}{\sqrt{x^{2} + 1}}$
  4. The derivative of a constant times a function is the constant times the derivative of the function.
    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:
      
      $\frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}}$
    So, the result is: $- \frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}}$
  The result is: $\frac{x}{\sqrt{x^{2} + 1}} - \frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}}$
The result of the chain rule is:

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

        2           x     
-1 - tan (x) + -----------
                  ________
                 /  2     
               \/  x  + 1 
--------------------------
      ________            
     /  2                 
   \/  x  + 1  - tan(x)

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

The graph

Derivative of y=ln(sqrt(x^2+1)-(2-1)*tgx)

Other calculators

How to use it?

Identical expressions

Similar expressions

Derivative of y=ln(sqrt(x^2+1)-(2-1)*tgx)

The graph:

Piecewise:

The solution