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

Derivative of tan(x^2)csc(x^2)

The solution

You have entered [src]

   / 2\    / 2\
tan\x /*csc\x /

$$\tan{\left(x^{2} \right)} \csc{\left(x^{2} \right)}$$

tan(x^2)*csc(x^2)

Detail solution

Apply the product rule:

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

$f{\left(x \right)} = \tan{\left(x^{2} \right)}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
1. Rewrite the function to be differentiated:
  
  $\tan{\left(x^{2} \right)} = \frac{\sin{\left(x^{2} \right)}}{\cos{\left(x^{2} \right)}}$
2. Apply the quotient rule, which is:
  
  $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
  
  $f{\left(x \right)} = \sin{\left(x^{2} \right)}$ and $g{\left(x \right)} = \cos{\left(x^{2} \right)}$ .
  
  To find $\frac{d}{d x} f{\left(x \right)}$ :
  1. Let $u = x^{2}$ .
  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 $\frac{d}{d x} x^{2}$ :
    1. Apply the power rule: $x^{2}$ goes to $2 x$
    The result of the chain rule is:
    
    $2 x \cos{\left(x^{2} \right)}$
  To find $\frac{d}{d x} g{\left(x \right)}$ :
  1. Let $u = x^{2}$ .
  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 $\frac{d}{d x} x^{2}$ :
    1. Apply the power rule: $x^{2}$ goes to $2 x$
    The result of the chain rule is:
    
    $- 2 x \sin{\left(x^{2} \right)}$
  Now plug in to the quotient rule:
  
  $\frac{2 x \sin^{2}{\left(x^{2} \right)} + 2 x \cos^{2}{\left(x^{2} \right)}}{\cos^{2}{\left(x^{2} \right)}}$
$g{\left(x \right)} = \csc{\left(x^{2} \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
1. Rewrite the function to be differentiated:
  
  $\csc{\left(x^{2} \right)} = \frac{1}{\sin{\left(x^{2} \right)}}$
2. Let $u = \sin{\left(x^{2} \right)}$ .
3. Apply the power rule: $\frac{1}{u}$ goes to $- \frac{1}{u^{2}}$
4. Then, apply the chain rule. Multiply by $\frac{d}{d x} \sin{\left(x^{2} \right)}$ :
  1. Let $u = x^{2}$ .
  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 $\frac{d}{d x} x^{2}$ :
    1. Apply the power rule: $x^{2}$ goes to $2 x$
    The result of the chain rule is:
    
    $2 x \cos{\left(x^{2} \right)}$
  The result of the chain rule is:
  
  $- \frac{2 x \cos{\left(x^{2} \right)}}{\sin^{2}{\left(x^{2} \right)}}$
The result is: $- \frac{2 x \cos{\left(x^{2} \right)} \tan{\left(x^{2} \right)}}{\sin^{2}{\left(x^{2} \right)}} + \frac{\left(2 x \sin^{2}{\left(x^{2} \right)} + 2 x \cos^{2}{\left(x^{2} \right)}\right) \csc{\left(x^{2} \right)}}{\cos^{2}{\left(x^{2} \right)}}$
Now simplify:

$\frac{2 x \sin{\left(x^{2} \right)}}{\cos^{2}{\left(x^{2} \right)}}$

The answer is:

$\frac{2 x \sin{\left(x^{2} \right)}}{\cos^{2}{\left(x^{2} \right)}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

    /       2/ 2\\    / 2\          / 2\    / 2\    / 2\
2*x*\1 + tan \x //*csc\x / - 2*x*cot\x /*csc\x /*tan\x /

$$2 x \left(\tan^{2}{\left(x^{2} \right)} + 1\right) \csc{\left(x^{2} \right)} - 2 x \tan{\left(x^{2} \right)} \cot{\left(x^{2} \right)} \csc{\left(x^{2} \right)}$$

Simplify

The second derivative [src]

  /       2/ 2\   /     / 2\      2    2/ 2\      2 /       2/ 2\\\    / 2\      2 /       2/ 2\\    / 2\      2 /       2/ 2\\    / 2\\    / 2\
2*\1 + tan \x / + \- cot\x / + 2*x *cot \x / + 2*x *\1 + cot \x ///*tan\x / - 4*x *\1 + tan \x //*cot\x / + 4*x *\1 + tan \x //*tan\x //*csc\x /

$$2 \left(4 x^{2} \left(\tan^{2}{\left(x^{2} \right)} + 1\right) \tan{\left(x^{2} \right)} - 4 x^{2} \left(\tan^{2}{\left(x^{2} \right)} + 1\right) \cot{\left(x^{2} \right)} + \left(2 x^{2} \left(\cot^{2}{\left(x^{2} \right)} + 1\right) + 2 x^{2} \cot^{2}{\left(x^{2} \right)} - \cot{\left(x^{2} \right)}\right) \tan{\left(x^{2} \right)} + \tan^{2}{\left(x^{2} \right)} + 1\right) \csc{\left(x^{2} \right)}$$

Simplify

The third derivative [src]

    /  /          2/ 2\      2    3/ 2\       2 /       2/ 2\\    / 2\\    / 2\     /       2/ 2\      2 /       2/ 2\\    / 2\\    / 2\     /       2/ 2\\ /     / 2\      2 /       2/ 2\\      2    2/ 2\\     /       2/ 2\\ /     / 2\      2    2/ 2\      2 /       2/ 2\\\\    / 2\
4*x*\- \-3 - 6*cot \x / + 2*x *cot \x / + 10*x *\1 + cot \x //*cot\x //*tan\x / - 3*\1 + tan \x / + 4*x *\1 + tan \x //*tan\x //*cot\x / + 2*\1 + tan \x //*\3*tan\x / + 2*x *\1 + tan \x // + 4*x *tan \x // + 3*\1 + tan \x //*\- cot\x / + 2*x *cot \x / + 2*x *\1 + cot \x ////*csc\x /

$$4 x \left(2 \left(\tan^{2}{\left(x^{2} \right)} + 1\right) \left(2 x^{2} \left(\tan^{2}{\left(x^{2} \right)} + 1\right) + 4 x^{2} \tan^{2}{\left(x^{2} \right)} + 3 \tan{\left(x^{2} \right)}\right) + 3 \left(\tan^{2}{\left(x^{2} \right)} + 1\right) \left(2 x^{2} \left(\cot^{2}{\left(x^{2} \right)} + 1\right) + 2 x^{2} \cot^{2}{\left(x^{2} \right)} - \cot{\left(x^{2} \right)}\right) - 3 \left(4 x^{2} \left(\tan^{2}{\left(x^{2} \right)} + 1\right) \tan{\left(x^{2} \right)} + \tan^{2}{\left(x^{2} \right)} + 1\right) \cot{\left(x^{2} \right)} - \left(10 x^{2} \left(\cot^{2}{\left(x^{2} \right)} + 1\right) \cot{\left(x^{2} \right)} + 2 x^{2} \cot^{3}{\left(x^{2} \right)} - 6 \cot^{2}{\left(x^{2} \right)} - 3\right) \tan{\left(x^{2} \right)}\right) \csc{\left(x^{2} \right)}$$

Simplify

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Derivative of tan(x^2)csc(x^2)

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The solution