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ln(sin(x- one / four *cos(x)))
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The derivative of the function/ ln(sin(x-1/4*cos(x)))

Derivative of ln(sin(x-1/4*cos(x)))

The solution

You have entered [src]

   /   /    cos(x)\\
log|sin|x - ------||
   \   \      4   //

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

log(sin(x - cos(x)/4))

Detail solution

Let $u = \sin{\left(x - \frac{\cos{\left(x \right)}}{4} \right)}$ .
The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
Then, apply the chain rule. Multiply by $\frac{d}{d x} \sin{\left(x - \frac{\cos{\left(x \right)}}{4} \right)}$ :
1. Let $u = x - \frac{\cos{\left(x \right)}}{4}$ .
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 $\frac{d}{d x} \left(x - \frac{\cos{\left(x \right)}}{4}\right)$ :
  1. Differentiate $x - \frac{\cos{\left(x \right)}}{4}$ 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 cosine is negative sine:
        
        $\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}$
      So, the result is: $\frac{\sin{\left(x \right)}}{4}$
    The result is: $\frac{\sin{\left(x \right)}}{4} + 1$
  The result of the chain rule is:
  
  $\left(\frac{\sin{\left(x \right)}}{4} + 1\right) \cos{\left(x - \frac{\cos{\left(x \right)}}{4} \right)}$
The result of the chain rule is:

$\frac{\left(\frac{\sin{\left(x \right)}}{4} + 1\right) \cos{\left(x - \frac{\cos{\left(x \right)}}{4} \right)}}{\sin{\left(x - \frac{\cos{\left(x \right)}}{4} \right)}}$
Now simplify:

$\frac{\frac{\sin{\left(x \right)}}{4} + 1}{\tan{\left(x - \frac{\cos{\left(x \right)}}{4} \right)}}$

The answer is:

$\frac{\frac{\sin{\left(x \right)}}{4} + 1}{\tan{\left(x - \frac{\cos{\left(x \right)}}{4} \right)}}$

The graph

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The first derivative [src]

/    sin(x)\    /    cos(x)\
|1 + ------|*cos|x - ------|
\      4   /    \      4   /
----------------------------
         /    cos(x)\       
      sin|x - ------|       
         \      4   /

$$\frac{\left(\frac{\sin{\left(x \right)}}{4} + 1\right) \cos{\left(x - \frac{\cos{\left(x \right)}}{4} \right)}}{\sin{\left(x - \frac{\cos{\left(x \right)}}{4} \right)}}$$

Simplify

The second derivative [src]

                              2    2/    cos(x)\               /    cos(x)\
                  (4 + sin(x)) *cos |x - ------|   4*cos(x)*cos|x - ------|
              2                     \      4   /               \      4   /
- (4 + sin(x))  - ------------------------------ + ------------------------
                            2/    cos(x)\                 /    cos(x)\     
                         sin |x - ------|              sin|x - ------|     
                             \      4   /                 \      4   /     
---------------------------------------------------------------------------
                                     16

$$\frac{- \left(\sin{\left(x \right)} + 4\right)^{2} - \frac{\left(\sin{\left(x \right)} + 4\right)^{2} \cos^{2}{\left(x - \frac{\cos{\left(x \right)}}{4} \right)}}{\sin^{2}{\left(x - \frac{\cos{\left(x \right)}}{4} \right)}} + \frac{4 \cos{\left(x \right)} \cos{\left(x - \frac{\cos{\left(x \right)}}{4} \right)}}{\sin{\left(x - \frac{\cos{\left(x \right)}}{4} \right)}}}{16}$$

Simplify

The third derivative [src]

                                     3    3/    cos(x)\               3    /    cos(x)\        /    cos(x)\               2/    cos(x)\                    
                         (4 + sin(x)) *cos |x - ------|   (4 + sin(x)) *cos|x - ------|   8*cos|x - ------|*sin(x)   6*cos |x - ------|*(4 + sin(x))*cos(x)
                                           \      4   /                    \      4   /        \      4   /                \      4   /                    
-6*(4 + sin(x))*cos(x) + ------------------------------ + ----------------------------- - ------------------------ - --------------------------------------
                                   3/    cos(x)\                    /    cos(x)\                 /    cos(x)\                      2/    cos(x)\           
                                sin |x - ------|                 sin|x - ------|              sin|x - ------|                   sin |x - ------|           
                                    \      4   /                    \      4   /                 \      4   /                       \      4   /           
-----------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                             32

$$\frac{\frac{\left(\sin{\left(x \right)} + 4\right)^{3} \cos{\left(x - \frac{\cos{\left(x \right)}}{4} \right)}}{\sin{\left(x - \frac{\cos{\left(x \right)}}{4} \right)}} + \frac{\left(\sin{\left(x \right)} + 4\right)^{3} \cos^{3}{\left(x - \frac{\cos{\left(x \right)}}{4} \right)}}{\sin^{3}{\left(x - \frac{\cos{\left(x \right)}}{4} \right)}} - 6 \left(\sin{\left(x \right)} + 4\right) \cos{\left(x \right)} - \frac{6 \left(\sin{\left(x \right)} + 4\right) \cos{\left(x \right)} \cos^{2}{\left(x - \frac{\cos{\left(x \right)}}{4} \right)}}{\sin^{2}{\left(x - \frac{\cos{\left(x \right)}}{4} \right)}} - \frac{8 \sin{\left(x \right)} \cos{\left(x - \frac{\cos{\left(x \right)}}{4} \right)}}{\sin{\left(x - \frac{\cos{\left(x \right)}}{4} \right)}}}{32}$$

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Derivative of ln(sin(x-1/4*cos(x)))

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