Derivative of y=sqrt(ln*sin(3x))

The solution

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  _________________
\/ log(x)*sin(3*x)

$$\sqrt{\log{\left(x \right)} \sin{\left(3 x \right)}}$$

d /  _________________\
--\\/ log(x)*sin(3*x) /
dx

$$\frac{d}{d x} \sqrt{\log{\left(x \right)} \sin{\left(3 x \right)}}$$

Transform

Detail solution

Let $u = \log{\left(x \right)} \sin{\left(3 x \right)}$ .
Apply the power rule: $\sqrt{u}$ goes to $\frac{1}{2 \sqrt{u}}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \log{\left(x \right)} \sin{\left(3 x \right)}$ :
1. Apply the product rule:
  
  $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
  
  $f{\left(x \right)} = \log{\left(x \right)}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
  1. The derivative of $\log{\left(x \right)}$ is $\frac{1}{x}$ .
  $g{\left(x \right)} = \sin{\left(3 x \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
  1. Let $u = 3 x$ .
  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 $\frac{d}{d x} 3 x$ :
    1. The derivative of a constant times a function is the constant times the derivative of the function.
      1. Apply the power rule: $x$ goes to $1$
      So, the result is: $3$
    The result of the chain rule is:
    
    $3 \cos{\left(3 x \right)}$
  The result is: $3 \log{\left(x \right)} \cos{\left(3 x \right)} + \frac{\sin{\left(3 x \right)}}{x}$
The result of the chain rule is:

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

$\frac{3 x \log{\left(x \right)} \cos{\left(3 x \right)} + \sin{\left(3 x \right)}}{2 x \sqrt{\log{\left(x \right)} \sin{\left(3 x \right)}}}$

The answer is:

$\frac{3 x \log{\left(x \right)} \cos{\left(3 x \right)} + \sin{\left(3 x \right)}}{2 x \sqrt{\log{\left(x \right)} \sin{\left(3 x \right)}}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

  _________________ /sin(3*x)   3*cos(3*x)*log(x)\
\/ log(x)*sin(3*x) *|-------- + -----------------|
                    \  2*x              2        /
--------------------------------------------------
                 log(x)*sin(3*x)

$$\frac{\sqrt{\log{\left(x \right)} \sin{\left(3 x \right)}} \left(\frac{3 \log{\left(x \right)} \cos{\left(3 x \right)}}{2} + \frac{\sin{\left(3 x \right)}}{2 x}\right)}{\log{\left(x \right)} \sin{\left(3 x \right)}}$$

Simplify

The second derivative [src]

                    /                                                                                                                                                     2\
                    |                                              /sin(3*x)                    \            sin(3*x)                       /sin(3*x)                    \ |
                    |                                            3*|-------- + 3*cos(3*x)*log(x)|*cos(3*x)   -------- + 3*cos(3*x)*log(x)   |-------- + 3*cos(3*x)*log(x)| |
  _________________ |3*cos(3*x)   9*log(x)*sin(3*x)   sin(3*x)     \   x                        /               x                           \   x                        / |
\/ log(x)*sin(3*x) *|---------- - ----------------- - -------- - ----------------------------------------- - ---------------------------- + -------------------------------|
                    |    x                2                2                     2*sin(3*x)                           2*x*log(x)                   4*log(x)*sin(3*x)       |
                    \                                   2*x                                                                                                                /
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                                                                              log(x)*sin(3*x)

$$\frac{\sqrt{\log{\left(x \right)} \sin{\left(3 x \right)}} \left(- \frac{9 \log{\left(x \right)} \sin{\left(3 x \right)}}{2} - \frac{3 \cdot \left(3 \log{\left(x \right)} \cos{\left(3 x \right)} + \frac{\sin{\left(3 x \right)}}{x}\right) \cos{\left(3 x \right)}}{2 \sin{\left(3 x \right)}} + \frac{\left(3 \log{\left(x \right)} \cos{\left(3 x \right)} + \frac{\sin{\left(3 x \right)}}{x}\right)^{2}}{4 \log{\left(x \right)} \sin{\left(3 x \right)}} + \frac{3 \cos{\left(3 x \right)}}{x} - \frac{3 \log{\left(x \right)} \cos{\left(3 x \right)} + \frac{\sin{\left(3 x \right)}}{x}}{2 x \log{\left(x \right)}} - \frac{\sin{\left(3 x \right)}}{2 x^{2}}\right)}{\log{\left(x \right)} \sin{\left(3 x \right)}}$$

Simplify

The third derivative [src]

                    /                                     sin(3*x)   6*cos(3*x)                                                                                       /sin(3*x)   6*cos(3*x)                    \                                                                                       3                                   2                                            2     /sin(3*x)                    \ /sin(3*x)   6*cos(3*x)                    \                                            \
                    |                                     -------- - ---------- + 9*log(x)*sin(3*x)   sin(3*x)                       sin(3*x)                       3*|-------- - ---------- + 9*log(x)*sin(3*x)|*cos(3*x)        2      /sin(3*x)                    \   /sin(3*x)                    \      /sin(3*x)                    \               /sin(3*x)                    \    3*|-------- + 3*cos(3*x)*log(x)|*|-------- - ---------- + 9*log(x)*sin(3*x)|     /sin(3*x)                    \         |
                    |                                         2          x                            -------- + 3*cos(3*x)*log(x)   -------- + 3*cos(3*x)*log(x)     |    2          x                         |            9*cos (3*x)*|-------- + 3*cos(3*x)*log(x)|   |-------- + 3*cos(3*x)*log(x)|    9*|-------- + 3*cos(3*x)*log(x)| *cos(3*x)   3*|-------- + 3*cos(3*x)*log(x)|      \   x                        / |    2          x                         |   3*|-------- + 3*cos(3*x)*log(x)|*cos(3*x)|
  _________________ |sin(3*x)   9*sin(3*x)   9*cos(3*x)      x                                           x                              x                             \   x                                     /                        \   x                        /   \   x                        /      \   x                        /               \   x                        /                                     \   x                                     /     \   x                        /         |
\/ log(x)*sin(3*x) *|-------- - ---------- - ---------- + ----------------------------------------- + ---------------------------- + ---------------------------- + ------------------------------------------------------ + ------------------------------------------ + ------------------------------- - ------------------------------------------ - --------------------------------- - ---------------------------------------------------------------------------- + -----------------------------------------|
                    |    3          x              2                       x*log(x)                             2    2                          2                                          sin(3*x)                                             2                                    2       2                                      2                                  2                                                  4*log(x)*sin(3*x)                                             x*log(x)*sin(3*x)            |
                    \   x                       2*x                                                            x *log (x)                    2*x *log(x)                                                                                     sin (3*x)                          8*log (x)*sin (3*x)                     4*log(x)*sin (3*x)                      4*x*log (x)*sin(3*x)                                                                                                                                 /
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                                                                                                                                                                                                                                                   log(x)*sin(3*x)

$$\frac{\sqrt{\log{\left(x \right)} \sin{\left(3 x \right)}} \left(\frac{9 \cdot \left(3 \log{\left(x \right)} \cos{\left(3 x \right)} + \frac{\sin{\left(3 x \right)}}{x}\right) \cos^{2}{\left(3 x \right)}}{\sin^{2}{\left(3 x \right)}} + \frac{3 \cdot \left(9 \log{\left(x \right)} \sin{\left(3 x \right)} - \frac{6 \cos{\left(3 x \right)}}{x} + \frac{\sin{\left(3 x \right)}}{x^{2}}\right) \cos{\left(3 x \right)}}{\sin{\left(3 x \right)}} - \frac{9 \left(3 \log{\left(x \right)} \cos{\left(3 x \right)} + \frac{\sin{\left(3 x \right)}}{x}\right)^{2} \cos{\left(3 x \right)}}{4 \log{\left(x \right)} \sin^{2}{\left(3 x \right)}} - \frac{3 \cdot \left(3 \log{\left(x \right)} \cos{\left(3 x \right)} + \frac{\sin{\left(3 x \right)}}{x}\right) \left(9 \log{\left(x \right)} \sin{\left(3 x \right)} - \frac{6 \cos{\left(3 x \right)}}{x} + \frac{\sin{\left(3 x \right)}}{x^{2}}\right)}{4 \log{\left(x \right)} \sin{\left(3 x \right)}} - \frac{9 \sin{\left(3 x \right)}}{x} + \frac{\left(3 \log{\left(x \right)} \cos{\left(3 x \right)} + \frac{\sin{\left(3 x \right)}}{x}\right)^{3}}{8 \log{\left(x \right)}^{2} \sin^{2}{\left(3 x \right)}} + \frac{3 \cdot \left(3 \log{\left(x \right)} \cos{\left(3 x \right)} + \frac{\sin{\left(3 x \right)}}{x}\right) \cos{\left(3 x \right)}}{x \log{\left(x \right)} \sin{\left(3 x \right)}} + \frac{9 \log{\left(x \right)} \sin{\left(3 x \right)} - \frac{6 \cos{\left(3 x \right)}}{x} + \frac{\sin{\left(3 x \right)}}{x^{2}}}{x \log{\left(x \right)}} - \frac{9 \cos{\left(3 x \right)}}{2 x^{2}} - \frac{3 \left(3 \log{\left(x \right)} \cos{\left(3 x \right)} + \frac{\sin{\left(3 x \right)}}{x}\right)^{2}}{4 x \log{\left(x \right)}^{2} \sin{\left(3 x \right)}} + \frac{3 \log{\left(x \right)} \cos{\left(3 x \right)} + \frac{\sin{\left(3 x \right)}}{x}}{2 x^{2} \log{\left(x \right)}} + \frac{\sin{\left(3 x \right)}}{x^{3}} + \frac{3 \log{\left(x \right)} \cos{\left(3 x \right)} + \frac{\sin{\left(3 x \right)}}{x}}{x^{2} \log{\left(x \right)}^{2}}\right)}{\log{\left(x \right)} \sin{\left(3 x \right)}}$$

Simplify

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Derivative of y=sqrt(ln*sin(3x))

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