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Derivative of:
Derivative of x^x^x
Derivative of x^-6
Derivative of y=3x²+5x+4
Derivative of sin(x)+3
Identical expressions
y=(ln5x)^sin(sqrt(4x- one))
y equally (ln5x) to the power of sinus of ( square root of (4x minus 1))
y equally (ln5x) to the power of sinus of ( square root of (4x minus one))
y=(ln5x)^sin(√(4x-1))
y=(ln5x)sin(sqrt(4x-1))
y=ln5xsinsqrt4x-1
y=ln5x^sinsqrt4x-1
Similar expressions
y=(ln5x)^sin(sqrt(4x+1))

The derivative of the function/ y=(ln5x)^sin(sqrt(4x-1))

Derivative of y=(ln5x)^sin(sqrt(4x-1))

The solution

You have entered [src]

             /  _________\
          sin\\/ 4*x - 1 /
(log(5*x))

\log{\left(5 x \right)}^{\sin{\left(\sqrt{4 x - 1} \right)}}

log(5*x)^sin(sqrt(4*x - 1))

Detail solution

Don't know the steps in finding this derivative.

But the derivative is

$\left(\log{\left(\sin{\left(\sqrt{4 x - 1} \right)} \right)} + 1\right) \sin^{\sin{\left(\sqrt{4 x - 1} \right)}}{\left(\sqrt{4 x - 1} \right)}$
Now simplify:

$\left(\log{\left(\sin{\left(\sqrt{4 x - 1} \right)} \right)} + 1\right) \sin^{\sin{\left(\sqrt{4 x - 1} \right)}}{\left(\sqrt{4 x - 1} \right)}$

The answer is:

\left(\log{\left(\sin{\left(\sqrt{4 x - 1} \right)} \right)} + 1\right) \sin^{\sin{\left(\sqrt{4 x - 1} \right)}}{\left(\sqrt{4 x - 1} \right)}

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

             /  _________\ /   /  _________\        /  _________\              \
          sin\\/ 4*x - 1 / |sin\\/ 4*x - 1 /   2*cos\\/ 4*x - 1 /*log(log(5*x))|
(log(5*x))                *|---------------- + --------------------------------|
                           |   x*log(5*x)                  _________           |
                           \                             \/ 4*x - 1            /

\left(\frac{2 \log{\left(\log{\left(5 x \right)} \right)} \cos{\left(\sqrt{4 x - 1} \right)}}{\sqrt{4 x - 1}} + \frac{\sin{\left(\sqrt{4 x - 1} \right)}}{x \log{\left(5 x \right)}}\right) \log{\left(5 x \right)}^{\sin{\left(\sqrt{4 x - 1} \right)}}

Simplify

The second derivative [src]

                            /                                                       2                                                                                                                                          \
             /  __________\ |/   /  __________\        /  __________\              \       /  __________\      /  __________\                      /  __________\        /  __________\                        /  __________\  |
          sin\\/ -1 + 4*x / ||sin\\/ -1 + 4*x /   2*cos\\/ -1 + 4*x /*log(log(5*x))|    sin\\/ -1 + 4*x /   sin\\/ -1 + 4*x /   4*log(log(5*x))*sin\\/ -1 + 4*x /   4*cos\\/ -1 + 4*x /*log(log(5*x))     4*cos\\/ -1 + 4*x /  |
(log(5*x))                 *||----------------- + ---------------------------------|  - ----------------- - ----------------- - --------------------------------- - --------------------------------- + -----------------------|
                            ||    x*log(5*x)                   __________          |        2                   2    2                       -1 + 4*x                                   3/2                 __________         |
                            \\                               \/ -1 + 4*x           /       x *log(5*x)         x *log (5*x)                                                   (-1 + 4*x)                x*\/ -1 + 4*x *log(5*x)/

\left(\left(\frac{2 \log{\left(\log{\left(5 x \right)} \right)} \cos{\left(\sqrt{4 x - 1} \right)}}{\sqrt{4 x - 1}} + \frac{\sin{\left(\sqrt{4 x - 1} \right)}}{x \log{\left(5 x \right)}}\right)^{2} - \frac{4 \log{\left(\log{\left(5 x \right)} \right)} \sin{\left(\sqrt{4 x - 1} \right)}}{4 x - 1} - \frac{4 \log{\left(\log{\left(5 x \right)} \right)} \cos{\left(\sqrt{4 x - 1} \right)}}{\left(4 x - 1\right)^{\frac{3}{2}}} + \frac{4 \cos{\left(\sqrt{4 x - 1} \right)}}{x \sqrt{4 x - 1} \log{\left(5 x \right)}} - \frac{\sin{\left(\sqrt{4 x - 1} \right)}}{x^{2} \log{\left(5 x \right)}} - \frac{\sin{\left(\sqrt{4 x - 1} \right)}}{x^{2} \log{\left(5 x \right)}^{2}}\right) \log{\left(5 x \right)}^{\sin{\left(\sqrt{4 x - 1} \right)}}

Simplify

The third derivative [src]

                            /                                                       3                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                \
             /  __________\ |/   /  __________\        /  __________\              \      /   /  __________\        /  __________\              \ /   /  __________\      /  __________\                      /  __________\        /  __________\                        /  __________\  \        /  __________\                      /  __________\        /  __________\        /  __________\                       /  __________\         /  __________\                        /  __________\           /  __________\            /  __________\              /  __________\   |
          sin\\/ -1 + 4*x / ||sin\\/ -1 + 4*x /   2*cos\\/ -1 + 4*x /*log(log(5*x))|      |sin\\/ -1 + 4*x /   2*cos\\/ -1 + 4*x /*log(log(5*x))| |sin\\/ -1 + 4*x /   sin\\/ -1 + 4*x /   4*log(log(5*x))*sin\\/ -1 + 4*x /   4*cos\\/ -1 + 4*x /*log(log(5*x))     4*cos\\/ -1 + 4*x /  |   8*cos\\/ -1 + 4*x /*log(log(5*x))   2*sin\\/ -1 + 4*x /   2*sin\\/ -1 + 4*x /   3*sin\\/ -1 + 4*x /   24*log(log(5*x))*sin\\/ -1 + 4*x /   24*cos\\/ -1 + 4*x /*log(log(5*x))    12*sin\\/ -1 + 4*x /     12*cos\\/ -1 + 4*x /       6*cos\\/ -1 + 4*x /         6*cos\\/ -1 + 4*x /   |
(log(5*x))                 *||----------------- + ---------------------------------|  - 3*|----------------- + ---------------------------------|*|----------------- + ----------------- + --------------------------------- + --------------------------------- - -----------------------| - --------------------------------- + ------------------- + ------------------- + ------------------- + ---------------------------------- + ---------------------------------- - --------------------- - ------------------------ - ------------------------ - -------------------------|
                            ||    x*log(5*x)                   __________          |      |    x*log(5*x)                   __________          | |    2                   2    2                       -1 + 4*x                                   3/2                 __________         |                       3/2                  3                     3    3                3    2                                2                                   5/2              x*(-1 + 4*x)*log(5*x)               3/2             2   __________             2   __________    2     |
                            \\                               \/ -1 + 4*x           /      \                               \/ -1 + 4*x           / \   x *log(5*x)         x *log (5*x)                                                   (-1 + 4*x)                x*\/ -1 + 4*x *log(5*x)/             (-1 + 4*x)                    x *log(5*x)           x *log (5*x)          x *log (5*x)                 (-1 + 4*x)                          (-1 + 4*x)                                         x*(-1 + 4*x)   *log(5*x)   x *\/ -1 + 4*x *log(5*x)   x *\/ -1 + 4*x *log (5*x)/

\left(\left(\frac{2 \log{\left(\log{\left(5 x \right)} \right)} \cos{\left(\sqrt{4 x - 1} \right)}}{\sqrt{4 x - 1}} + \frac{\sin{\left(\sqrt{4 x - 1} \right)}}{x \log{\left(5 x \right)}}\right)^{3} - 3 \left(\frac{2 \log{\left(\log{\left(5 x \right)} \right)} \cos{\left(\sqrt{4 x - 1} \right)}}{\sqrt{4 x - 1}} + \frac{\sin{\left(\sqrt{4 x - 1} \right)}}{x \log{\left(5 x \right)}}\right) \left(\frac{4 \log{\left(\log{\left(5 x \right)} \right)} \sin{\left(\sqrt{4 x - 1} \right)}}{4 x - 1} + \frac{4 \log{\left(\log{\left(5 x \right)} \right)} \cos{\left(\sqrt{4 x - 1} \right)}}{\left(4 x - 1\right)^{\frac{3}{2}}} - \frac{4 \cos{\left(\sqrt{4 x - 1} \right)}}{x \sqrt{4 x - 1} \log{\left(5 x \right)}} + \frac{\sin{\left(\sqrt{4 x - 1} \right)}}{x^{2} \log{\left(5 x \right)}} + \frac{\sin{\left(\sqrt{4 x - 1} \right)}}{x^{2} \log{\left(5 x \right)}^{2}}\right) + \frac{24 \log{\left(\log{\left(5 x \right)} \right)} \sin{\left(\sqrt{4 x - 1} \right)}}{\left(4 x - 1\right)^{2}} - \frac{8 \log{\left(\log{\left(5 x \right)} \right)} \cos{\left(\sqrt{4 x - 1} \right)}}{\left(4 x - 1\right)^{\frac{3}{2}}} + \frac{24 \log{\left(\log{\left(5 x \right)} \right)} \cos{\left(\sqrt{4 x - 1} \right)}}{\left(4 x - 1\right)^{\frac{5}{2}}} - \frac{12 \sin{\left(\sqrt{4 x - 1} \right)}}{x \left(4 x - 1\right) \log{\left(5 x \right)}} - \frac{12 \cos{\left(\sqrt{4 x - 1} \right)}}{x \left(4 x - 1\right)^{\frac{3}{2}} \log{\left(5 x \right)}} - \frac{6 \cos{\left(\sqrt{4 x - 1} \right)}}{x^{2} \sqrt{4 x - 1} \log{\left(5 x \right)}} - \frac{6 \cos{\left(\sqrt{4 x - 1} \right)}}{x^{2} \sqrt{4 x - 1} \log{\left(5 x \right)}^{2}} + \frac{2 \sin{\left(\sqrt{4 x - 1} \right)}}{x^{3} \log{\left(5 x \right)}} + \frac{3 \sin{\left(\sqrt{4 x - 1} \right)}}{x^{3} \log{\left(5 x \right)}^{2}} + \frac{2 \sin{\left(\sqrt{4 x - 1} \right)}}{x^{3} \log{\left(5 x \right)}^{3}}\right) \log{\left(5 x \right)}^{\sin{\left(\sqrt{4 x - 1} \right)}}

Simplify

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Derivative of y=(ln5x)^sin(sqrt(4x-1))

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The solution