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Identical expressions
tan^ two (x^ four - two)
tangent of squared (x to the power of 4 minus 2)
tangent of to the power of two (x to the power of four minus two)
tan2(x4-2)
tan2x4-2
tan²(x⁴-2)
tan to the power of 2(x to the power of 4-2)
tan^2x^4-2
Similar expressions
tan^2(x^4+2)

The derivative of the function/ tan^2(x^4-2)

Derivative of tan^2(x^4-2)

The solution

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   2/ 4    \
tan \x  - 2/

$$\tan^{2}{\left(x^{4} - 2 \right)}$$

tan(x^4 - 2)^2

Detail solution

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

$\frac{2 \left(4 x^{3} \sin^{2}{\left(x^{4} - 2 \right)} + 4 x^{3} \cos^{2}{\left(x^{4} - 2 \right)}\right) \tan{\left(x^{4} - 2 \right)}}{\cos^{2}{\left(x^{4} - 2 \right)}}$
Now simplify:

$\frac{8 x^{3} \tan{\left(x^{4} - 2 \right)}}{\cos^{2}{\left(x^{4} - 2 \right)}}$

The answer is:

$\frac{8 x^{3} \tan{\left(x^{4} - 2 \right)}}{\cos^{2}{\left(x^{4} - 2 \right)}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

   3 /       2/ 4    \\    / 4    \
8*x *\1 + tan \x  - 2//*tan\x  - 2/

$$8 x^{3} \left(\tan^{2}{\left(x^{4} - 2 \right)} + 1\right) \tan{\left(x^{4} - 2 \right)}$$

Simplify

The second derivative [src]

   2 /       2/      4\\ /     /      4\      4 /       2/      4\\      4    2/      4\\
8*x *\1 + tan \-2 + x //*\3*tan\-2 + x / + 4*x *\1 + tan \-2 + x // + 8*x *tan \-2 + x //

$$8 x^{2} \left(\tan^{2}{\left(x^{4} - 2 \right)} + 1\right) \left(4 x^{4} \left(\tan^{2}{\left(x^{4} - 2 \right)} + 1\right) + 8 x^{4} \tan^{2}{\left(x^{4} - 2 \right)} + 3 \tan{\left(x^{4} - 2 \right)}\right)$$

Simplify

The third derivative [src]

     /       2/      4\\ /     /      4\       4 /       2/      4\\       8    3/      4\       4    2/      4\       8 /       2/      4\\    /      4\\
16*x*\1 + tan \-2 + x //*\3*tan\-2 + x / + 18*x *\1 + tan \-2 + x // + 32*x *tan \-2 + x / + 36*x *tan \-2 + x / + 64*x *\1 + tan \-2 + x //*tan\-2 + x //

$$16 x \left(\tan^{2}{\left(x^{4} - 2 \right)} + 1\right) \left(64 x^{8} \left(\tan^{2}{\left(x^{4} - 2 \right)} + 1\right) \tan{\left(x^{4} - 2 \right)} + 32 x^{8} \tan^{3}{\left(x^{4} - 2 \right)} + 18 x^{4} \left(\tan^{2}{\left(x^{4} - 2 \right)} + 1\right) + 36 x^{4} \tan^{2}{\left(x^{4} - 2 \right)} + 3 \tan{\left(x^{4} - 2 \right)}\right)$$

Simplify

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Derivative of tan^2(x^4-2)

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The solution