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Derivative of:
Derivative of x^3*lnx
Derivative of sin(3)^(2)*x
Derivative of (8*x-15)^5
Derivative of (3*x-2)^7
Identical expressions
(log two *x)^(cosx^2)
( logarithm of 2 multiply by x) to the power of ( co sinus of e of x squared )
( logarithm of two multiply by x) to the power of ( co sinus of e of x squared )
(log2*x)(cosx2)
log2*xcosx2
(log2*x)^(cosx²)
(log2*x) to the power of (cosx to the power of 2)
(log2x)^(cosx^2)
(log2x)(cosx2)
log2xcosx2
log2x^cosx^2

The derivative of the function/ (log2*x)^(cosx^2)

Derivative of (log2*x)^(cosx^2)

The solution

You have entered [src]

             2   
          cos (x)
(log(2*x))

$$\log{\left(2 x \right)}^{\cos^{2}{\left(x \right)}}$$

  /             2   \
d |          cos (x)|
--\(log(2*x))       /
dx

$$\frac{d}{d x} \log{\left(2 x \right)}^{\cos^{2}{\left(x \right)}}$$

Transform

Detail solution

Don't know the steps in finding this derivative.

But the derivative is

$\left(\log{\left(\cos^{2}{\left(x \right)} \right)} + 1\right) \left|{\cos{\left(x \right)}}\right|^{2 \cos^{2}{\left(x \right)}}$

The answer is:

$\left(\log{\left(\cos^{2}{\left(x \right)} \right)} + 1\right) \left|{\cos{\left(x \right)}}\right|^{2 \cos^{2}{\left(x \right)}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

$$\left(- 2 \log{\left(\log{\left(2 x \right)} \right)} \sin{\left(x \right)} \cos{\left(x \right)} + \frac{\cos^{2}{\left(x \right)}}{x \log{\left(2 x \right)}}\right) \log{\left(2 x \right)}^{\cos^{2}{\left(x \right)}}$$

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

The graph

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Identical expressions

Derivative of (log2*x)^(cosx^2)

The graph:

Piecewise:

The solution