Mister Exam

Other calculators

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

Function f() - derivative -N order at the point
v

The graph:

from to

Piecewise:

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
  1. Let .

  2. The derivative of is .

  3. Then, apply the chain rule. Multiply by :

    1. Let .

    2. The derivative of sine is cosine:

    3. Then, apply the chain rule. Multiply by :

      1. Differentiate term by term:

        1. Apply the power rule: goes to

        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:

          So, the result is:

        The result is:

      The result of the chain rule is:

    The result of the chain rule is:

  4. Now simplify:


The answer is:

The graph
The first derivative [src]
/    sin(x)\    /    cos(x)\
|1 + ------|*cos|x - ------|
\      4   /    \      4   /
----------------------------
         /    cos(x)\       
      sin|x - ------|       
         \      4   /       
$$\frac{\left(\frac{\sin{\left(x \right)}}{4} + 1\right) \cos{\left(x - \frac{\cos{\left(x \right)}}{4} \right)}}{\sin{\left(x - \frac{\cos{\left(x \right)}}{4} \right)}}$$
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                                    
$$\frac{- \left(\sin{\left(x \right)} + 4\right)^{2} - \frac{\left(\sin{\left(x \right)} + 4\right)^{2} \cos^{2}{\left(x - \frac{\cos{\left(x \right)}}{4} \right)}}{\sin^{2}{\left(x - \frac{\cos{\left(x \right)}}{4} \right)}} + \frac{4 \cos{\left(x \right)} \cos{\left(x - \frac{\cos{\left(x \right)}}{4} \right)}}{\sin{\left(x - \frac{\cos{\left(x \right)}}{4} \right)}}}{16}$$
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                                                                            
$$\frac{\frac{\left(\sin{\left(x \right)} + 4\right)^{3} \cos{\left(x - \frac{\cos{\left(x \right)}}{4} \right)}}{\sin{\left(x - \frac{\cos{\left(x \right)}}{4} \right)}} + \frac{\left(\sin{\left(x \right)} + 4\right)^{3} \cos^{3}{\left(x - \frac{\cos{\left(x \right)}}{4} \right)}}{\sin^{3}{\left(x - \frac{\cos{\left(x \right)}}{4} \right)}} - 6 \left(\sin{\left(x \right)} + 4\right) \cos{\left(x \right)} - \frac{6 \left(\sin{\left(x \right)} + 4\right) \cos{\left(x \right)} \cos^{2}{\left(x - \frac{\cos{\left(x \right)}}{4} \right)}}{\sin^{2}{\left(x - \frac{\cos{\left(x \right)}}{4} \right)}} - \frac{8 \sin{\left(x \right)} \cos{\left(x - \frac{\cos{\left(x \right)}}{4} \right)}}{\sin{\left(x - \frac{\cos{\left(x \right)}}{4} \right)}}}{32}$$