Derivative of tan(x^8)

The solution

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   / 8\
tan\x /

$$\tan{\left(x^{8} \right)}$$

tan(x^8)

Detail solution

Rewrite the function to be differentiated:

$\tan{\left(x^{8} \right)} = \frac{\sin{\left(x^{8} \right)}}{\cos{\left(x^{8} \right)}}$
Apply the quotient rule, which is:

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

$f{\left(x \right)} = \sin{\left(x^{8} \right)}$ and $g{\left(x \right)} = \cos{\left(x^{8} \right)}$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Let $u = x^{8}$ .
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 $\frac{d}{d x} x^{8}$ :
  1. Apply the power rule: $x^{8}$ goes to $8 x^{7}$
  The result of the chain rule is:
  
  $8 x^{7} \cos{\left(x^{8} \right)}$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Let $u = x^{8}$ .
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 $\frac{d}{d x} x^{8}$ :
  1. Apply the power rule: $x^{8}$ goes to $8 x^{7}$
  The result of the chain rule is:
  
  $- 8 x^{7} \sin{\left(x^{8} \right)}$
Now plug in to the quotient rule:

$\frac{8 x^{7} \sin^{2}{\left(x^{8} \right)} + 8 x^{7} \cos^{2}{\left(x^{8} \right)}}{\cos^{2}{\left(x^{8} \right)}}$
Now simplify:

$\frac{8 x^{7}}{\cos^{2}{\left(x^{8} \right)}}$

The answer is:

$\frac{8 x^{7}}{\cos^{2}{\left(x^{8} \right)}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

   7 /       2/ 8\\
8*x *\1 + tan \x //

$$8 x^{7} \left(\tan^{2}{\left(x^{8} \right)} + 1\right)$$

Simplify

The second derivative [src]

   6 /       2/ 8\\ /        8    / 8\\
8*x *\1 + tan \x //*\7 + 16*x *tan\x //

$$8 x^{6} \left(16 x^{8} \tan{\left(x^{8} \right)} + 7\right) \left(\tan^{2}{\left(x^{8} \right)} + 1\right)$$

Simplify

8-я производная [src]

    /                                                2                               4                                3                                                                                                                                                                                                                                                                           2                                         4                                                                                 2                                        3                                         2                                         3                                        2                                         3                                          2         \
    |             2/ 8\             16 /       2/ 8\\               48 /       2/ 8\\                32 /       2/ 8\\             8 /       2/ 8\\    / 8\             56    7/ 8\ /       2/ 8\\              16    2/ 8\ /       2/ 8\\              48    6/ 8\ /       2/ 8\\              24    3/ 8\ /       2/ 8\\              40    5/ 8\ /       2/ 8\\               56 /       2/ 8\\     5/ 8\               56 /       2/ 8\\     / 8\               32    4/ 8\ /       2/ 8\\               24 /       2/ 8\\     / 8\               56 /       2/ 8\\     3/ 8\               48 /       2/ 8\\     4/ 8\               40 /       2/ 8\\     / 8\               32 /       2/ 8\\     2/ 8\               48 /       2/ 8\\     2/ 8\                40 /       2/ 8\\     3/ 8\|
128*\315 + 315*tan \x / + 73170720*x  *\1 + tan \x //  + 873463808*x  *\1 + tan \x //  + 1324646400*x  *\1 + tan \x //  + 4053420*x *\1 + tan \x //*tan\x / + 16777216*x  *tan \x /*\1 + tan \x // + 146341440*x  *tan \x /*\1 + tan \x // + 205520896*x  *tan \x /*\1 + tan \x // + 803631360*x  *tan \x /*\1 + tan \x // + 828506112*x  *tan \x /*\1 + tan \x // + 1006632960*x  *\1 + tan \x // *tan \x / + 1040187392*x  *\1 + tan \x // *tan\x / + 1324646400*x  *tan \x /*\1 + tan \x // + 1607262720*x  *\1 + tan \x // *tan\x / + 3221225472*x  *\1 + tan \x // *tan \x / + 5857345536*x  *\1 + tan \x // *tan \x / + 7042301952*x  *\1 + tan \x // *tan\x / + 7285555200*x  *\1 + tan \x // *tan \x / + 9248440320*x  *\1 + tan \x // *tan \x / + 10770579456*x  *\1 + tan \x // *tan \x //

$$128 \left(1040187392 x^{56} \left(\tan^{2}{\left(x^{8} \right)} + 1\right)^{4} \tan{\left(x^{8} \right)} + 3221225472 x^{56} \left(\tan^{2}{\left(x^{8} \right)} + 1\right)^{3} \tan^{3}{\left(x^{8} \right)} + 1006632960 x^{56} \left(\tan^{2}{\left(x^{8} \right)} + 1\right)^{2} \tan^{5}{\left(x^{8} \right)} + 16777216 x^{56} \left(\tan^{2}{\left(x^{8} \right)} + 1\right) \tan^{7}{\left(x^{8} \right)} + 873463808 x^{48} \left(\tan^{2}{\left(x^{8} \right)} + 1\right)^{4} + 9248440320 x^{48} \left(\tan^{2}{\left(x^{8} \right)} + 1\right)^{3} \tan^{2}{\left(x^{8} \right)} + 5857345536 x^{48} \left(\tan^{2}{\left(x^{8} \right)} + 1\right)^{2} \tan^{4}{\left(x^{8} \right)} + 205520896 x^{48} \left(\tan^{2}{\left(x^{8} \right)} + 1\right) \tan^{6}{\left(x^{8} \right)} + 7042301952 x^{40} \left(\tan^{2}{\left(x^{8} \right)} + 1\right)^{3} \tan{\left(x^{8} \right)} + 10770579456 x^{40} \left(\tan^{2}{\left(x^{8} \right)} + 1\right)^{2} \tan^{3}{\left(x^{8} \right)} + 828506112 x^{40} \left(\tan^{2}{\left(x^{8} \right)} + 1\right) \tan^{5}{\left(x^{8} \right)} + 1324646400 x^{32} \left(\tan^{2}{\left(x^{8} \right)} + 1\right)^{3} + 7285555200 x^{32} \left(\tan^{2}{\left(x^{8} \right)} + 1\right)^{2} \tan^{2}{\left(x^{8} \right)} + 1324646400 x^{32} \left(\tan^{2}{\left(x^{8} \right)} + 1\right) \tan^{4}{\left(x^{8} \right)} + 1607262720 x^{24} \left(\tan^{2}{\left(x^{8} \right)} + 1\right)^{2} \tan{\left(x^{8} \right)} + 803631360 x^{24} \left(\tan^{2}{\left(x^{8} \right)} + 1\right) \tan^{3}{\left(x^{8} \right)} + 73170720 x^{16} \left(\tan^{2}{\left(x^{8} \right)} + 1\right)^{2} + 146341440 x^{16} \left(\tan^{2}{\left(x^{8} \right)} + 1\right) \tan^{2}{\left(x^{8} \right)} + 4053420 x^{8} \left(\tan^{2}{\left(x^{8} \right)} + 1\right) \tan{\left(x^{8} \right)} + 315 \tan^{2}{\left(x^{8} \right)} + 315\right)$$

Simplify

The third derivative [src]

    5 /       2/ 8\\ /         16 /       2/ 8\\        16    2/ 8\        8    / 8\\
16*x *\1 + tan \x //*\21 + 64*x  *\1 + tan \x // + 128*x  *tan \x / + 168*x *tan\x //

$$16 x^{5} \left(\tan^{2}{\left(x^{8} \right)} + 1\right) \left(64 x^{16} \left(\tan^{2}{\left(x^{8} \right)} + 1\right) + 128 x^{16} \tan^{2}{\left(x^{8} \right)} + 168 x^{8} \tan{\left(x^{8} \right)} + 21\right)$$

Simplify

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Identical expressions

Derivative of tan(x^8)

The graph:

Piecewise:

The solution