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(x+ two ^x)^ln(x)
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Similar expressions
(x-2^x)^ln(x)

The derivative of the function/ (x+2^x)^ln(x)

Derivative of (x+2^x)^ln(x)

The solution

You have entered [src]

        log(x)
/     x\      
\x + 2 /

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

(x + 2^x)^log(x)

Detail solution

Don't know the steps in finding this derivative.

But the derivative is

$\left(\log{\left(\log{\left(x \right)} \right)} + 1\right) \log{\left(x \right)}^{\log{\left(x \right)}}$

The answer is:

\left(\log{\left(\log{\left(x \right)} \right)} + 1\right) \log{\left(x \right)}^{\log{\left(x \right)}}

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

        log(x) /   /     x\   /     x       \       \
/     x\       |log\x + 2 /   \1 + 2 *log(2)/*log(x)|
\x + 2 /      *|----------- + ----------------------|
               |     x                     x        |
               \                      x + 2         /

\left(2^{x} + x\right)^{\log{\left(x \right)}} \left(\frac{\left(2^{x} \log{\left(2 \right)} + 1\right) \log{\left(x \right)}}{2^{x} + x} + \frac{\log{\left(2^{x} + x \right)}}{x}\right)

Simplify

The second derivative [src]

               /                                      2                                2                                               \
        log(x) |/   /     x\   /     x       \       \       /     x\   /     x       \             /     x       \    x    2          |
/     x\       ||log\x + 2 /   \1 + 2 *log(2)/*log(x)|    log\x + 2 /   \1 + 2 *log(2)/ *log(x)   2*\1 + 2 *log(2)/   2 *log (2)*log(x)|
\x + 2 /      *||----------- + ----------------------|  - ----------- - ----------------------- + ----------------- + -----------------|
               ||     x                     x        |          2                      2                /     x\                 x     |
               |\                      x + 2         /         x               /     x\               x*\x + 2 /            x + 2      |
               \                                                               \x + 2 /                                                /

\left(2^{x} + x\right)^{\log{\left(x \right)}} \left(\frac{2^{x} \log{\left(2 \right)}^{2} \log{\left(x \right)}}{2^{x} + x} + \left(\frac{\left(2^{x} \log{\left(2 \right)} + 1\right) \log{\left(x \right)}}{2^{x} + x} + \frac{\log{\left(2^{x} + x \right)}}{x}\right)^{2} - \frac{\left(2^{x} \log{\left(2 \right)} + 1\right)^{2} \log{\left(x \right)}}{\left(2^{x} + x\right)^{2}} + \frac{2 \left(2^{x} \log{\left(2 \right)} + 1\right)}{x \left(2^{x} + x\right)} - \frac{\log{\left(2^{x} + x \right)}}{x^{2}}\right)

Simplify

The third derivative [src]

               /                                      3                                                            /                               2                                               \                    2                                        3                                                                                \
        log(x) |/   /     x\   /     x       \       \         /     x\     /   /     x\   /     x       \       \ |     /     x\   /     x       \             /     x       \    x    2          |     /     x       \      /     x       \     /     x       \            x    3                x    2         x    2    /     x       \       |
/     x\       ||log\x + 2 /   \1 + 2 *log(2)/*log(x)|    2*log\x + 2 /     |log\x + 2 /   \1 + 2 *log(2)/*log(x)| |  log\x + 2 /   \1 + 2 *log(2)/ *log(x)   2*\1 + 2 *log(2)/   2 *log (2)*log(x)|   3*\1 + 2 *log(2)/    3*\1 + 2 *log(2)/   2*\1 + 2 *log(2)/ *log(x)   2 *log (2)*log(x)   3*2 *log (2)   3*2 *log (2)*\1 + 2 *log(2)/*log(x)|
\x + 2 /      *||----------- + ----------------------|  + ------------- + 3*|----------- + ----------------------|*|- ----------- - ----------------------- + ----------------- + -----------------| - ------------------ - ----------------- + ------------------------- + ----------------- + ------------ - -----------------------------------|
               ||     x                     x        |           3          |     x                     x        | |        2                      2                /     x\                 x     |                2           2 /     x\                      3                      x           /     x\                         2             |
               |\                      x + 2         /          x           \                      x + 2         / |       x               /     x\               x*\x + 2 /            x + 2      |        /     x\           x *\x + 2 /              /     x\                  x + 2          x*\x + 2 /                 /     x\              |
               \                                                                                                   \                       \x + 2 /                                                /      x*\x + 2 /                                    \x + 2 /                                                            \x + 2 /              /

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

Simplify

The graph

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Identical expressions

Similar expressions

Derivative of (x+2^x)^ln(x)

The graph:

Piecewise:

The solution