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Similar expressions
y=sin^2((1+lnx)/x)

The derivative of the function/ y=sin^2((1-lnx)/x)

Derivative of y=sin^2((1-lnx)/x)

The solution

You have entered [src]

   2/1 - log(x)\
sin |----------|
    \    x     /

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

sin((1 - log(x))/x)^2

Detail solution

Let $u = \sin{\left(\frac{1 - \log{\left(x \right)}}{x} \right)}$ .
Apply the power rule: $u^{2}$ goes to $2 u$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \sin{\left(\frac{1 - \log{\left(x \right)}}{x} \right)}$ :
1. Let $u = \frac{1 - \log{\left(x \right)}}{x}$ .
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 $\frac{d}{d x} \frac{1 - \log{\left(x \right)}}{x}$ :
  1. Apply the quotient rule, which is:
    
    $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
    
    $f{\left(x \right)} = 1 - \log{\left(x \right)}$ and $g{\left(x \right)} = x$ .
    
    To find $\frac{d}{d x} f{\left(x \right)}$ :
    1. Differentiate $1 - \log{\left(x \right)}$ term by term:
      1. The derivative of the constant $1$ is zero.
      2. The derivative of a constant times a function is the constant times the derivative of the function.
        
        The derivative of $\log{\left(x \right)}$ is $\frac{1}{x}$ .
        
        So, the result is: $- \frac{1}{x}$
      The result is: $- \frac{1}{x}$
    To find $\frac{d}{d x} g{\left(x \right)}$ :
    1. Apply the power rule: $x$ goes to $1$
    Now plug in to the quotient rule:
    
    $\frac{\log{\left(x \right)} - 2}{x^{2}}$
  The result of the chain rule is:
  
  $\frac{\left(\log{\left(x \right)} - 2\right) \cos{\left(\frac{1 - \log{\left(x \right)}}{x} \right)}}{x^{2}}$
The result of the chain rule is:

$\frac{2 \left(\log{\left(x \right)} - 2\right) \sin{\left(\frac{1 - \log{\left(x \right)}}{x} \right)} \cos{\left(\frac{1 - \log{\left(x \right)}}{x} \right)}}{x^{2}}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

  /  1    1 - log(x)\    /1 - log(x)\    /1 - log(x)\
2*|- -- - ----------|*cos|----------|*sin|----------|
  |   2        2    |    \    x     /    \    x     /
  \  x        x     /

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

  /                                                            2/-1 + log(x)\                                      2/-1 + log(x)\                                                3    /-1 + log(x)\    /-1 + log(x)\\
  |                                                       3*cos |-----------|*(-5 + 2*log(x))*(-2 + log(x))   3*sin |-----------|*(-5 + 2*log(x))*(-2 + log(x))   4*(-2 + log(x)) *cos|-----------|*sin|-----------||
  |                      /-1 + log(x)\    /-1 + log(x)\         \     x     /                                       \     x     /                                                     \     x     /    \     x     /|
2*|- (-17 + 6*log(x))*cos|-----------|*sin|-----------| - ------------------------------------------------- + ------------------------------------------------- + --------------------------------------------------|
  |                      \     x     /    \     x     /                           x                                                   x                                                    2                        |
  \                                                                                                                                                                                       x                         /
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                                           4                                                                                                         
                                                                                                          x

\frac{2 \left(- \left(6 \log{\left(x \right)} - 17\right) \sin{\left(\frac{\log{\left(x \right)} - 1}{x} \right)} \cos{\left(\frac{\log{\left(x \right)} - 1}{x} \right)} + \frac{3 \left(\log{\left(x \right)} - 2\right) \left(2 \log{\left(x \right)} - 5\right) \sin^{2}{\left(\frac{\log{\left(x \right)} - 1}{x} \right)}}{x} - \frac{3 \left(\log{\left(x \right)} - 2\right) \left(2 \log{\left(x \right)} - 5\right) \cos^{2}{\left(\frac{\log{\left(x \right)} - 1}{x} \right)}}{x} + \frac{4 \left(\log{\left(x \right)} - 2\right)^{3} \sin{\left(\frac{\log{\left(x \right)} - 1}{x} \right)} \cos{\left(\frac{\log{\left(x \right)} - 1}{x} \right)}}{x^{2}}\right)}{x^{4}}

Simplify

The graph

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Derivative of y=sin^2((1-lnx)/x)

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The solution