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y=tg^ two (x^ three + one)
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y=tg2(x3+1)
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y=tg to the power of 2(x to the power of 3+1)
y=tg^2x^3+1
Similar expressions
y=tg^2(x^3-1)

The derivative of the function/ y=tg^2(x^3+1)

Derivative of y=tg^2(x^3+1)

The solution

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   2/ 3    \
tan \x  + 1/

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

d /   2/ 3    \\
--\tan \x  + 1//
dx

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

Transform

Detail solution

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

$\frac{2 \cdot \left(3 x^{2} \sin^{2}{\left(x^{3} + 1 \right)} + 3 x^{2} \cos^{2}{\left(x^{3} + 1 \right)}\right) \tan{\left(x^{3} + 1 \right)}}{\cos^{2}{\left(x^{3} + 1 \right)}}$
Now simplify:

$\frac{6 x^{2} \tan{\left(x^{3} + 1 \right)}}{\cos^{2}{\left(x^{3} + 1 \right)}}$

The answer is:

$\frac{6 x^{2} \tan{\left(x^{3} + 1 \right)}}{\cos^{2}{\left(x^{3} + 1 \right)}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

   2 /       2/ 3    \\    / 3    \
6*x *\1 + tan \x  + 1//*tan\x  + 1/

$$6 x^{2} \left(\tan^{2}{\left(x^{3} + 1 \right)} + 1\right) \tan{\left(x^{3} + 1 \right)}$$

Simplify

The second derivative [src]

    /       2/     3\\ /     /     3\      3 /       2/     3\\      3    2/     3\\
6*x*\1 + tan \1 + x //*\2*tan\1 + x / + 3*x *\1 + tan \1 + x // + 6*x *tan \1 + x //

$$6 x \left(\tan^{2}{\left(x^{3} + 1 \right)} + 1\right) \left(3 x^{3} \left(\tan^{2}{\left(x^{3} + 1 \right)} + 1\right) + 6 x^{3} \tan^{2}{\left(x^{3} + 1 \right)} + 2 \tan{\left(x^{3} + 1 \right)}\right)$$

Simplify

The third derivative [src]

   /       2/     3\\ /   3 /       2/     3\\       3    2/     3\       6    3/     3\       6 /       2/     3\\    /     3\      /     3\\
12*\1 + tan \1 + x //*\9*x *\1 + tan \1 + x // + 18*x *tan \1 + x / + 18*x *tan \1 + x / + 36*x *\1 + tan \1 + x //*tan\1 + x / + tan\1 + x //

$$12 \left(\tan^{2}{\left(x^{3} + 1 \right)} + 1\right) \left(36 x^{6} \left(\tan^{2}{\left(x^{3} + 1 \right)} + 1\right) \tan{\left(x^{3} + 1 \right)} + 18 x^{6} \tan^{3}{\left(x^{3} + 1 \right)} + 9 x^{3} \left(\tan^{2}{\left(x^{3} + 1 \right)} + 1\right) + 18 x^{3} \tan^{2}{\left(x^{3} + 1 \right)} + \tan{\left(x^{3} + 1 \right)}\right)$$

Simplify

The graph

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Derivative of y=tg^2(x^3+1)

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The solution