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tan(x)^ three +sin(3x^ two)
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tan(x)^3-sin(3x^2)

The derivative of the function/ tan(x)^3+sin(3x^2)

Derivative of tan(x)^3+sin(3x^2)

The solution

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   3         /   2\
tan (x) + sin\3*x /

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

d /   3         /   2\\
--\tan (x) + sin\3*x //
dx

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

Transform

Detail solution

Differentiate $\sin{\left(3 x^{2} \right)} + \tan^{3}{\left(x \right)}$ term by term:
1. Let $u = \tan{\left(x \right)}$ .
2. Apply the power rule: $u^{3}$ goes to $3 u^{2}$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \tan{\left(x \right)}$ :
  1. Rewrite the function to be differentiated:
    
    $\tan{\left(x \right)} = \frac{\sin{\left(x \right)}}{\cos{\left(x \right)}}$
  2. Apply the quotient rule, which is:
    
    $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
    
    $f{\left(x \right)} = \sin{\left(x \right)}$ and $g{\left(x \right)} = \cos{\left(x \right)}$ .
    
    To find $\frac{d}{d x} f{\left(x \right)}$ :
    1. The derivative of sine is cosine:
      
      $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
    To find $\frac{d}{d x} g{\left(x \right)}$ :
    1. The derivative of cosine is negative sine:
      
      $\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}$
    Now plug in to the quotient rule:
    
    $\frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}}$
  The result of the chain rule is:
  
  $\frac{3 \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \tan^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}}$
4. Let $u = 3 x^{2}$ .
5. The derivative of sine is cosine:
  
  $\frac{d}{d u} \sin{\left(u \right)} = \cos{\left(u \right)}$
6. Then, apply the chain rule. Multiply by $\frac{d}{d x} 3 x^{2}$ :
  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: $6 x$
  The result of the chain rule is:
  
  $6 x \cos{\left(3 x^{2} \right)}$
The result is: $6 x \cos{\left(3 x^{2} \right)} + \frac{3 \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \tan^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

   2    /         2   \          /   2\
tan (x)*\3 + 3*tan (x)/ + 6*x*cos\3*x /

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

Simplify

The second derivative [src]

  /             2                                                            \
  |/       2   \              3    /       2   \      2    /   2\      /   2\|
6*\\1 + tan (x)/ *tan(x) + tan (x)*\1 + tan (x)/ - 6*x *sin\3*x / + cos\3*x //

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

Simplify

The third derivative [src]

  /             3                                                                               2        \
  |/       2   \        3    /   2\           /   2\        4    /       2   \     /       2   \     2   |
6*\\1 + tan (x)/  - 36*x *cos\3*x / - 18*x*sin\3*x / + 2*tan (x)*\1 + tan (x)/ + 7*\1 + tan (x)/ *tan (x)/

$$6 \left(- 36 x^{3} \cos{\left(3 x^{2} \right)} - 18 x \sin{\left(3 x^{2} \right)} + \left(\tan^{2}{\left(x \right)} + 1\right)^{3} + 7 \left(\tan^{2}{\left(x \right)} + 1\right)^{2} \tan^{2}{\left(x \right)} + 2 \left(\tan^{2}{\left(x \right)} + 1\right) \tan^{4}{\left(x \right)}\right)$$

Simplify

The graph

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Derivative of tan(x)^3+sin(3x^2)

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The solution