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

Derivative of tan(2x^2-1)/sin(2x^2-1)

The solution

You have entered [src]

   /   2    \
tan\2*x  - 1/
-------------
   /   2    \
sin\2*x  - 1/

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

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

Detail solution

Apply the quotient rule, which is:

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

$f{\left(x \right)} = \tan{\left(2 x^{2} - 1 \right)}$ and $g{\left(x \right)} = \sin{\left(2 x^{2} - 1 \right)}$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Rewrite the function to be differentiated:
  
  $\tan{\left(2 x^{2} - 1 \right)} = \frac{\sin{\left(2 x^{2} - 1 \right)}}{\cos{\left(2 x^{2} - 1 \right)}}$
2. Apply the quotient rule, which is:
  
  $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
  
  $f{\left(x \right)} = \sin{\left(2 x^{2} - 1 \right)}$ and $g{\left(x \right)} = \cos{\left(2 x^{2} - 1 \right)}$ .
  
  To find $\frac{d}{d x} f{\left(x \right)}$ :
  1. Let $u = 2 x^{2} - 1$ .
  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 $\frac{d}{d x} \left(2 x^{2} - 1\right)$ :
    1. Differentiate $2 x^{2} - 1$ term by term:
      1. The derivative of a constant times a function is the constant times the derivative of the function.
        
        Apply the power rule: $x^{2}$ goes to $2 x$
        
        So, the result is: $4 x$
      2. The derivative of the constant $-1$ is zero.
      The result is: $4 x$
    The result of the chain rule is:
    
    $4 x \cos{\left(2 x^{2} - 1 \right)}$
  To find $\frac{d}{d x} g{\left(x \right)}$ :
  1. Let $u = 2 x^{2} - 1$ .
  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 $\frac{d}{d x} \left(2 x^{2} - 1\right)$ :
    1. Differentiate $2 x^{2} - 1$ term by term:
      1. The derivative of a constant times a function is the constant times the derivative of the function.
        
        Apply the power rule: $x^{2}$ goes to $2 x$
        
        So, the result is: $4 x$
      2. The derivative of the constant $-1$ is zero.
      The result is: $4 x$
    The result of the chain rule is:
    
    $- 4 x \sin{\left(2 x^{2} - 1 \right)}$
  Now plug in to the quotient rule:
  
  $\frac{4 x \sin^{2}{\left(2 x^{2} - 1 \right)} + 4 x \cos^{2}{\left(2 x^{2} - 1 \right)}}{\cos^{2}{\left(2 x^{2} - 1 \right)}}$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Let $u = 2 x^{2} - 1$ .
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 $\frac{d}{d x} \left(2 x^{2} - 1\right)$ :
  1. Differentiate $2 x^{2} - 1$ term by term:
    1. The derivative of a constant times a function is the constant times the derivative of the function.
      1. Apply the power rule: $x^{2}$ goes to $2 x$
      So, the result is: $4 x$
    2. The derivative of the constant $-1$ is zero.
    The result is: $4 x$
  The result of the chain rule is:
  
  $4 x \cos{\left(2 x^{2} - 1 \right)}$
Now plug in to the quotient rule:

$\frac{- 4 x \cos{\left(2 x^{2} - 1 \right)} \tan{\left(2 x^{2} - 1 \right)} + \frac{\left(4 x \sin^{2}{\left(2 x^{2} - 1 \right)} + 4 x \cos^{2}{\left(2 x^{2} - 1 \right)}\right) \sin{\left(2 x^{2} - 1 \right)}}{\cos^{2}{\left(2 x^{2} - 1 \right)}}}{\sin^{2}{\left(2 x^{2} - 1 \right)}}$
Now simplify:

$\frac{4 x \left(\sin{\left(2 x^{2} - 1 \right)} - \cos^{3}{\left(2 x^{2} - 1 \right)} \tan{\left(2 x^{2} - 1 \right)}\right)}{\sin^{2}{\left(2 x^{2} - 1 \right)} \cos^{2}{\left(2 x^{2} - 1 \right)}}$

The answer is:

$\frac{4 x \left(\sin{\left(2 x^{2} - 1 \right)} - \cos^{3}{\left(2 x^{2} - 1 \right)} \tan{\left(2 x^{2} - 1 \right)}\right)}{\sin^{2}{\left(2 x^{2} - 1 \right)} \cos^{2}{\left(2 x^{2} - 1 \right)}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

    /       2/   2    \\          /   2    \    /   2    \
4*x*\1 + tan \2*x  - 1//   4*x*cos\2*x  - 1/*tan\2*x  - 1/
------------------------ - -------------------------------
        /   2    \                     2/   2    \        
     sin\2*x  - 1/                  sin \2*x  - 1/

$$\frac{4 x \left(\tan^{2}{\left(2 x^{2} - 1 \right)} + 1\right)}{\sin{\left(2 x^{2} - 1 \right)}} - \frac{4 x \cos{\left(2 x^{2} - 1 \right)} \tan{\left(2 x^{2} - 1 \right)}}{\sin^{2}{\left(2 x^{2} - 1 \right)}}$$

Simplify

The second derivative [src]

  /                      /          /        2\      2    2/        2\\                                                                 2 /       2/        2\\    /        2\\
  |       2/        2\   |   2   cos\-1 + 2*x /   8*x *cos \-1 + 2*x /|    /        2\      2 /       2/        2\\    /        2\   8*x *\1 + tan \-1 + 2*x //*cos\-1 + 2*x /|
4*|1 + tan \-1 + 2*x / + |4*x  - -------------- + --------------------|*tan\-1 + 2*x / + 8*x *\1 + tan \-1 + 2*x //*tan\-1 + 2*x / - -----------------------------------------|
  |                      |          /        2\        2/        2\   |                                                                               /        2\             |
  \                      \       sin\-1 + 2*x /     sin \-1 + 2*x /   /                                                                            sin\-1 + 2*x /             /
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                    /        2\                                                                                
                                                                                 sin\-1 + 2*x /

$$\frac{4 \left(8 x^{2} \left(\tan^{2}{\left(2 x^{2} - 1 \right)} + 1\right) \tan{\left(2 x^{2} - 1 \right)} - \frac{8 x^{2} \left(\tan^{2}{\left(2 x^{2} - 1 \right)} + 1\right) \cos{\left(2 x^{2} - 1 \right)}}{\sin{\left(2 x^{2} - 1 \right)}} + \left(4 x^{2} + \frac{8 x^{2} \cos^{2}{\left(2 x^{2} - 1 \right)}}{\sin^{2}{\left(2 x^{2} - 1 \right)}} - \frac{\cos{\left(2 x^{2} - 1 \right)}}{\sin{\left(2 x^{2} - 1 \right)}}\right) \tan{\left(2 x^{2} - 1 \right)} + \tan^{2}{\left(2 x^{2} - 1 \right)} + 1\right)}{\sin{\left(2 x^{2} - 1 \right)}}$$

Simplify

The third derivative [src]

     /  /          2/        2\       2    /        2\       2    3/        2\\                                                                                                                                           /          /        2\      2    2/        2\\     /       2/        2\      2 /       2/        2\\    /        2\\    /        2\\
     |  |     6*cos \-1 + 2*x /   20*x *cos\-1 + 2*x /   24*x *cos \-1 + 2*x /|    /        2\     /       2/        2\\ /     /        2\      2 /       2/        2\\      2    2/        2\\     /       2/        2\\ |   2   cos\-1 + 2*x /   8*x *cos \-1 + 2*x /|   3*\1 + tan \-1 + 2*x / + 8*x *\1 + tan \-1 + 2*x //*tan\-1 + 2*x //*cos\-1 + 2*x /|
16*x*|- |-3 - ----------------- + -------------------- + ---------------------|*tan\-1 + 2*x / + 2*\1 + tan \-1 + 2*x //*\3*tan\-1 + 2*x / + 4*x *\1 + tan \-1 + 2*x // + 8*x *tan \-1 + 2*x // + 3*\1 + tan \-1 + 2*x //*|4*x  - -------------- + --------------------| - ----------------------------------------------------------------------------------|
     |  |         2/        2\          /        2\            3/        2\   |                                                                                                                                           |          /        2\        2/        2\   |                                        /        2\                                  |
     \  \      sin \-1 + 2*x /       sin\-1 + 2*x /         sin \-1 + 2*x /   /                                                                                                                                           \       sin\-1 + 2*x /     sin \-1 + 2*x /   /                                     sin\-1 + 2*x /                                  /
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                                                                                                           /        2\                                                                                                                                                                        
                                                                                                                                                                        sin\-1 + 2*x /

$$\frac{16 x \left(3 \left(\tan^{2}{\left(2 x^{2} - 1 \right)} + 1\right) \left(4 x^{2} + \frac{8 x^{2} \cos^{2}{\left(2 x^{2} - 1 \right)}}{\sin^{2}{\left(2 x^{2} - 1 \right)}} - \frac{\cos{\left(2 x^{2} - 1 \right)}}{\sin{\left(2 x^{2} - 1 \right)}}\right) + 2 \left(\tan^{2}{\left(2 x^{2} - 1 \right)} + 1\right) \left(4 x^{2} \left(\tan^{2}{\left(2 x^{2} - 1 \right)} + 1\right) + 8 x^{2} \tan^{2}{\left(2 x^{2} - 1 \right)} + 3 \tan{\left(2 x^{2} - 1 \right)}\right) - \frac{3 \left(8 x^{2} \left(\tan^{2}{\left(2 x^{2} - 1 \right)} + 1\right) \tan{\left(2 x^{2} - 1 \right)} + \tan^{2}{\left(2 x^{2} - 1 \right)} + 1\right) \cos{\left(2 x^{2} - 1 \right)}}{\sin{\left(2 x^{2} - 1 \right)}} - \left(\frac{20 x^{2} \cos{\left(2 x^{2} - 1 \right)}}{\sin{\left(2 x^{2} - 1 \right)}} + \frac{24 x^{2} \cos^{3}{\left(2 x^{2} - 1 \right)}}{\sin^{3}{\left(2 x^{2} - 1 \right)}} - 3 - \frac{6 \cos^{2}{\left(2 x^{2} - 1 \right)}}{\sin^{2}{\left(2 x^{2} - 1 \right)}}\right) \tan{\left(2 x^{2} - 1 \right)}\right)}{\sin{\left(2 x^{2} - 1 \right)}}$$

Simplify

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Derivative of tan(2x^2-1)/sin(2x^2-1)

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