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Identical expressions
sin^ six (x-lnx)
sinus of to the power of 6(x minus lnx)
sinus of to the power of six (x minus lnx)
sin6(x-lnx)
sin6x-lnx
sin⁶(x-lnx)
sin^6x-lnx
Similar expressions
sin^6(x+lnx)

The derivative of the function/ sin^6(x-lnx)

Derivative of sin^6(x-lnx)

The solution

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   6            
sin (x - log(x))

$$\sin^{6}{\left(x - \log{\left(x \right)} \right)}$$

sin(x - log(x))^6

Detail solution

Let $u = \sin{\left(x - \log{\left(x \right)} \right)}$ .
Apply the power rule: $u^{6}$ goes to $6 u^{5}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \sin{\left(x - \log{\left(x \right)} \right)}$ :
1. Let $u = x - \log{\left(x \right)}$ .
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 $\frac{d}{d x} \left(x - \log{\left(x \right)}\right)$ :
  1. Differentiate $x - \log{\left(x \right)}$ term by term:
    1. Apply the power rule: $x$ goes to $1$
    2. The derivative of a constant times a function is the constant times the derivative of the function.
      1. The derivative of $\log{\left(x \right)}$ is $\frac{1}{x}$ .
      So, the result is: $- \frac{1}{x}$
    The result is: $1 - \frac{1}{x}$
  The result of the chain rule is:
  
  $\left(1 - \frac{1}{x}\right) \cos{\left(x - \log{\left(x \right)} \right)}$
The result of the chain rule is:

$6 \left(1 - \frac{1}{x}\right) \sin^{5}{\left(x - \log{\left(x \right)} \right)} \cos{\left(x - \log{\left(x \right)} \right)}$
Now simplify:

$\frac{6 \left(x - 1\right) \sin^{5}{\left(x - \log{\left(x \right)} \right)} \cos{\left(x - \log{\left(x \right)} \right)}}{x}$

The answer is:

$\frac{6 \left(x - 1\right) \sin^{5}{\left(x - \log{\left(x \right)} \right)} \cos{\left(x - \log{\left(x \right)} \right)}}{x}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

     5             /    1\                
6*sin (x - log(x))*|1 - -|*cos(x - log(x))
                   \    x/

$$6 \left(1 - \frac{1}{x}\right) \sin^{5}{\left(x - \log{\left(x \right)} \right)} \cos{\left(x - \log{\left(x \right)} \right)}$$

Simplify

The second derivative [src]

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

$$6 \left(- \left(1 - \frac{1}{x}\right)^{2} \sin^{2}{\left(x - \log{\left(x \right)} \right)} + 5 \left(1 - \frac{1}{x}\right)^{2} \cos^{2}{\left(x - \log{\left(x \right)} \right)} + \frac{\sin{\left(x - \log{\left(x \right)} \right)} \cos{\left(x - \log{\left(x \right)} \right)}}{x^{2}}\right) \sin^{4}{\left(x - \log{\left(x \right)} \right)}$$

Simplify

The third derivative [src]

                   /                                                                                   3             /    1\                                              2             /    1\                \
                   |          3                              3                                    3*sin (x - log(x))*|1 - -|        2                               15*cos (x - log(x))*|1 - -|*sin(x - log(x))|
     3             |   /    1\     3                  /    1\     2                                                  \    x/   2*sin (x - log(x))*cos(x - log(x))                       \    x/                |
6*sin (x - log(x))*|20*|1 - -| *cos (x - log(x)) - 16*|1 - -| *sin (x - log(x))*cos(x - log(x)) - -------------------------- - ---------------------------------- + -------------------------------------------|
                   |   \    x/                        \    x/                                                  2                                3                                         2                    |
                   \                                                                                          x                                x                                         x                     /

$$6 \left(- 16 \left(1 - \frac{1}{x}\right)^{3} \sin^{2}{\left(x - \log{\left(x \right)} \right)} \cos{\left(x - \log{\left(x \right)} \right)} + 20 \left(1 - \frac{1}{x}\right)^{3} \cos^{3}{\left(x - \log{\left(x \right)} \right)} - \frac{3 \left(1 - \frac{1}{x}\right) \sin^{3}{\left(x - \log{\left(x \right)} \right)}}{x^{2}} + \frac{15 \left(1 - \frac{1}{x}\right) \sin{\left(x - \log{\left(x \right)} \right)} \cos^{2}{\left(x - \log{\left(x \right)} \right)}}{x^{2}} - \frac{2 \sin^{2}{\left(x - \log{\left(x \right)} \right)} \cos{\left(x - \log{\left(x \right)} \right)}}{x^{3}}\right) \sin^{3}{\left(x - \log{\left(x \right)} \right)}$$

Simplify

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Derivative of sin^6(x-lnx)

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The solution