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Derivative of:
Derivative of e^(6*x)
Derivative of -1/y
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Derivative of log(1+1/(x^2))
Identical expressions
y=√[tg(3x^ two)]
y equally √[tg(3x squared )]
y equally √[tg(3x to the power of two)]
y=√[tg(3x2)]
y=√[tg3x2]
y=√[tg(3x²)]
y=√[tg(3x to the power of 2)]
y=√[tg3x^2]

The derivative of the function/ y=√[tg(3x^2)]

Derivative of y=√[tg(3x^2)]

The solution

You have entered [src]

   ___________
  /    /   2\ 
\/  tan\3*x /

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

  /   ___________\
d |  /    /   2\ |
--\\/  tan\3*x / /
dx

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

Transform

Detail solution

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

$\frac{6 x \sin^{2}{\left(3 x^{2} \right)} + 6 x \cos^{2}{\left(3 x^{2} \right)}}{2 \cos^{2}{\left(3 x^{2} \right)} \sqrt{\tan{\left(3 x^{2} \right)}}}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

    /       2/   2\\
3*x*\1 + tan \3*x //
--------------------
      ___________   
     /    /   2\    
   \/  tan\3*x /

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

Simplify

The second derivative [src]

                   /                          ___________      2 /       2/   2\\\
  /       2/   2\\ |      1              2   /    /   2\    3*x *\1 + tan \3*x //|
3*\1 + tan \3*x //*|-------------- + 12*x *\/  tan\3*x /  - ---------------------|
                   |   ___________                                  3/2/   2\    |
                   |  /    /   2\                                tan   \3*x /    |
                   \\/  tan\3*x /                                                /

$$3 \left(\tan^{2}{\left(3 x^{2} \right)} + 1\right) \left(- \frac{3 x^{2} \left(\tan^{2}{\left(3 x^{2} \right)} + 1\right)}{\tan^{\frac{3}{2}}{\left(3 x^{2} \right)}} + 12 x^{2} \sqrt{\tan{\left(3 x^{2} \right)}} + \frac{1}{\sqrt{\tan{\left(3 x^{2} \right)}}}\right)$$

Simplify

The third derivative [src]

                      /                                                                                                      2\
                      |     ___________          2/   2\                           2 /       2/   2\\      2 /       2/   2\\ |
     /       2/   2\\ |    /    /   2\    1 + tan \3*x /       2    3/2/   2\   4*x *\1 + tan \3*x //   3*x *\1 + tan \3*x // |
27*x*\1 + tan \3*x //*|4*\/  tan\3*x /  - -------------- + 16*x *tan   \3*x / - --------------------- + ----------------------|
                      |                       3/2/   2\                                ___________              5/2/   2\     |
                      |                    tan   \3*x /                               /    /   2\            tan   \3*x /     |
                      \                                                             \/  tan\3*x /                             /

$$27 x \left(\tan^{2}{\left(3 x^{2} \right)} + 1\right) \left(\frac{3 x^{2} \left(\tan^{2}{\left(3 x^{2} \right)} + 1\right)^{2}}{\tan^{\frac{5}{2}}{\left(3 x^{2} \right)}} - \frac{4 x^{2} \left(\tan^{2}{\left(3 x^{2} \right)} + 1\right)}{\sqrt{\tan{\left(3 x^{2} \right)}}} + 16 x^{2} \tan^{\frac{3}{2}}{\left(3 x^{2} \right)} - \frac{\tan^{2}{\left(3 x^{2} \right)} + 1}{\tan^{\frac{3}{2}}{\left(3 x^{2} \right)}} + 4 \sqrt{\tan{\left(3 x^{2} \right)}}\right)$$

Simplify

The graph

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Identical expressions

Derivative of y=√[tg(3x^2)]

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The solution