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Derivative of:
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Derivative of y=x/lnx
Identical expressions
y=(sin(3x))^sqrt(x)
y equally ( sinus of (3x)) to the power of square root of (x)
y=(sin(3x))^√(x)
y=(sin(3x))sqrt(x)
y=sin3xsqrtx
y=sin3x^sqrtx

The derivative of the function/ y=(sin(3x))^sqrt(x)

Derivative of y=(sin(3x))^sqrt(x)

The solution

You have entered [src]

            ___
          \/ x 
(sin(3*x))

\sin^{\sqrt{x}}{\left(3 x \right)}

  /            ___\
d |          \/ x |
--\(sin(3*x))     /
dx

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

Transform

Detail solution

Don't know the steps in finding this derivative.

But the derivative is

$x^{\frac{\sqrt{x}}{2}} \left(\log{\left(\sqrt{x} \right)} + 1\right)$
Now simplify:

$\frac{x^{\frac{\sqrt{x}}{2}} \left(\log{\left(x \right)} + 2\right)}{2}$

The answer is:

\frac{x^{\frac{\sqrt{x}}{2}} \left(\log{\left(x \right)} + 2\right)}{2}

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

                /                                              3                                                                                                                                                                                                         \
                |            /                    ___         \      /                    ___         \ /                                                 ___    2     \                                                                                                 |
                |            |log(sin(3*x))   6*\/ x *cos(3*x)|      |log(sin(3*x))   6*\/ x *cos(3*x)| |     ___   log(sin(3*x))    12*cos(3*x)     36*\/ x *cos (3*x)|                                                                                                 |
                |            |------------- + ----------------|    3*|------------- + ----------------|*|36*\/ x  + ------------- - -------------- + ------------------|                                                                                                 |
            ___ |            |      ___           sin(3*x)    |      |      ___           sin(3*x)    | |                 3/2         ___                   2          |                          ___    3             ___                     2                         |
          \/ x  |     27     \    \/ x                        /      \    \/ x                        / \                x          \/ x *sin(3*x)       sin (3*x)     /   3*log(sin(3*x))   54*\/ x *cos (3*x)   54*\/ x *cos(3*x)      27*cos (3*x)        9*cos(3*x)  |
(sin(3*x))     *|- ------- + ----------------------------------- - ----------------------------------------------------------------------------------------------------- + --------------- + ------------------ + ----------------- - ----------------- - ---------------|
                |      ___                    8                                                                      8                                                             5/2              3                  sin(3*x)           ___    2           3/2         |
                \  2*\/ x                                                                                                                                                       8*x              sin (3*x)                            2*\/ x *sin (3*x)   4*x   *sin(3*x)/

\left(\frac{54 \sqrt{x} \cos{\left(3 x \right)}}{\sin{\left(3 x \right)}} + \frac{54 \sqrt{x} \cos^{3}{\left(3 x \right)}}{\sin^{3}{\left(3 x \right)}} + \frac{\left(\frac{6 \sqrt{x} \cos{\left(3 x \right)}}{\sin{\left(3 x \right)}} + \frac{\log{\left(\sin{\left(3 x \right)} \right)}}{\sqrt{x}}\right)^{3}}{8} - \frac{3 \cdot \left(\frac{6 \sqrt{x} \cos{\left(3 x \right)}}{\sin{\left(3 x \right)}} + \frac{\log{\left(\sin{\left(3 x \right)} \right)}}{\sqrt{x}}\right) \left(36 \sqrt{x} + \frac{36 \sqrt{x} \cos^{2}{\left(3 x \right)}}{\sin^{2}{\left(3 x \right)}} - \frac{12 \cos{\left(3 x \right)}}{\sqrt{x} \sin{\left(3 x \right)}} + \frac{\log{\left(\sin{\left(3 x \right)} \right)}}{x^{\frac{3}{2}}}\right)}{8} - \frac{27}{2 \sqrt{x}} - \frac{27 \cos^{2}{\left(3 x \right)}}{2 \sqrt{x} \sin^{2}{\left(3 x \right)}} - \frac{9 \cos{\left(3 x \right)}}{4 x^{\frac{3}{2}} \sin{\left(3 x \right)}} + \frac{3 \log{\left(\sin{\left(3 x \right)} \right)}}{8 x^{\frac{5}{2}}}\right) \sin^{\sqrt{x}}{\left(3 x \right)}

Simplify

The graph

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Identical expressions

Derivative of y=(sin(3x))^sqrt(x)

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The solution