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Derivative of:
Derivative of e^(3*x)
Derivative of sin(y)
Derivative of x^log(x)
Derivative of cos(9x)
Identical expressions
sqrt(log(7x^ three -3x))
square root of ( logarithm of (7x cubed minus 3x))
square root of ( logarithm of (7x to the power of three minus 3x))
√(log(7x^3-3x))
sqrt(log(7x3-3x))
sqrtlog7x3-3x
sqrt(log(7x³-3x))
sqrt(log(7x to the power of 3-3x))
sqrtlog7x^3-3x
Similar expressions
sqrt(log(7x^3+3x))

The derivative of the function/ sqrt(log(7x^3-3x))

Derivative of sqrt(log(7x^3-3x))

The solution

You have entered [src]

   _________________
  /    /   3      \ 
\/  log\7*x  - 3*x/

$$\sqrt{\log{\left(7 x^{3} - 3 x \right)}}$$

  /   _________________\
d |  /    /   3      \ |
--\\/  log\7*x  - 3*x/ /
dx

$$\frac{d}{d x} \sqrt{\log{\left(7 x^{3} - 3 x \right)}}$$

Transform

Detail solution

Let $u = \log{\left(7 x^{3} - 3 x \right)}$ .
Apply the power rule: $\sqrt{u}$ goes to $\frac{1}{2 \sqrt{u}}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \log{\left(7 x^{3} - 3 x \right)}$ :
1. Let $u = 7 x^{3} - 3 x$ .
2. The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(7 x^{3} - 3 x\right)$ :
  1. Differentiate $7 x^{3} - 3 x$ term by term:
    1. The derivative of a constant times a function is the constant times the derivative of the function.
      1. Apply the power rule: $x^{3}$ goes to $3 x^{2}$
      So, the result is: $21 x^{2}$
    2. The derivative of a constant times a function is the constant times the derivative of the function.
      1. The derivative of a constant times a function is the constant times the derivative of the function.
        
        Apply the power rule: $x$ goes to $1$
        
        So, the result is: $3$
      So, the result is: $-3$
    The result is: $21 x^{2} - 3$
  The result of the chain rule is:
  
  $\frac{21 x^{2} - 3}{7 x^{3} - 3 x}$
The result of the chain rule is:

$\frac{21 x^{2} - 3}{2 \cdot \left(7 x^{3} - 3 x\right) \sqrt{\log{\left(7 x^{3} - 3 x \right)}}}$
Now simplify:

$\frac{3 \cdot \left(7 x^{2} - 1\right)}{2 x \left(7 x^{2} - 3\right) \sqrt{\log{\left(x \left(7 x^{2} - 3\right) \right)}}}$

The answer is:

$\frac{3 \cdot \left(7 x^{2} - 1\right)}{2 x \left(7 x^{2} - 3\right) \sqrt{\log{\left(x \left(7 x^{2} - 3\right) \right)}}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

                      2            
             -3 + 21*x             
-----------------------------------
                  _________________
  /   3      \   /    /   3      \ 
2*\7*x  - 3*x/*\/  log\7*x  - 3*x/

$$\frac{21 x^{2} - 3}{2 \cdot \left(7 x^{3} - 3 x\right) \sqrt{\log{\left(7 x^{3} - 3 x \right)}}}$$

Simplify

The second derivative [src]

  /                  2                            2          \
  |       /        2\                  /        2\           |
  |     3*\-1 + 7*x /                3*\-1 + 7*x /           |
3*|7 - ---------------- - -----------------------------------|
  |       2 /        2\      2 /        2\    /  /        2\\|
  \    2*x *\-3 + 7*x /   4*x *\-3 + 7*x /*log\x*\-3 + 7*x ///
--------------------------------------------------------------
                            ____________________              
             /        2\   /    /  /        2\\               
             \-3 + 7*x /*\/  log\x*\-3 + 7*x //

$$\frac{3 \cdot \left(7 - \frac{3 \left(7 x^{2} - 1\right)^{2}}{2 x^{2} \cdot \left(7 x^{2} - 3\right)} - \frac{3 \left(7 x^{2} - 1\right)^{2}}{4 x^{2} \cdot \left(7 x^{2} - 3\right) \log{\left(x \left(7 x^{2} - 3\right) \right)}}\right)}{\left(7 x^{2} - 3\right) \sqrt{\log{\left(x \left(7 x^{2} - 3\right) \right)}}}$$

Simplify

The third derivative [src]

  /                                   3                                                              3                                       3           \
  |       /        2\      /        2\                /        2\                         /        2\                             /        2\            |
  |    63*\-1 + 7*x /    9*\-1 + 7*x /             63*\-1 + 7*x /                      27*\-1 + 7*x /                          27*\-1 + 7*x /            |
3*|7 - -------------- + --------------- - -------------------------------- + ------------------------------------ + -------------------------------------|
  |              2                    2     /        2\    /  /        2\\                   2                                      2                    |
  |      -3 + 7*x        2 /        2\    2*\-3 + 7*x /*log\x*\-3 + 7*x //      2 /        2\     /  /        2\\      2 /        2\     2/  /        2\\|
  \                     x *\-3 + 7*x /                                       4*x *\-3 + 7*x / *log\x*\-3 + 7*x //   8*x *\-3 + 7*x / *log \x*\-3 + 7*x ///
----------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                           ____________________                                                           
                                                            /        2\   /    /  /        2\\                                                            
                                                          x*\-3 + 7*x /*\/  log\x*\-3 + 7*x //

$$\frac{3 \cdot \left(7 - \frac{63 \cdot \left(7 x^{2} - 1\right)}{7 x^{2} - 3} - \frac{63 \cdot \left(7 x^{2} - 1\right)}{2 \cdot \left(7 x^{2} - 3\right) \log{\left(x \left(7 x^{2} - 3\right) \right)}} + \frac{9 \left(7 x^{2} - 1\right)^{3}}{x^{2} \left(7 x^{2} - 3\right)^{2}} + \frac{27 \left(7 x^{2} - 1\right)^{3}}{4 x^{2} \left(7 x^{2} - 3\right)^{2} \log{\left(x \left(7 x^{2} - 3\right) \right)}} + \frac{27 \left(7 x^{2} - 1\right)^{3}}{8 x^{2} \left(7 x^{2} - 3\right)^{2} \log{\left(x \left(7 x^{2} - 3\right) \right)}^{2}}\right)}{x \left(7 x^{2} - 3\right) \sqrt{\log{\left(x \left(7 x^{2} - 3\right) \right)}}}$$

Simplify

The graph

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Derivative of sqrt(log(7x^3-3x))

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Piecewise:

The solution