Mister Exam

Other calculators


log(x*sin(x)+cos(x))

Derivative of log(x*sin(x)+cos(x))

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

The graph:

from to

Piecewise:

The solution

You have entered [src]
log(x*sin(x) + cos(x))
$$\log{\left(x \sin{\left(x \right)} + \cos{\left(x \right)} \right)}$$
log(x*sin(x) + cos(x))
Detail solution
  1. Let .

  2. The derivative of is .

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

    1. Differentiate term by term:

      1. Apply the product rule:

        ; to find :

        1. Apply the power rule: goes to

        ; to find :

        1. The derivative of sine is cosine:

        The result is:

      2. The derivative of cosine is negative sine:

      The result is:

    The result of the chain rule is:


The answer is:

The graph
The first derivative [src]
     x*cos(x)    
-----------------
x*sin(x) + cos(x)
$$\frac{x \cos{\left(x \right)}}{x \sin{\left(x \right)} + \cos{\left(x \right)}}$$
The second derivative [src]
                 2    2               
                x *cos (x)            
-x*sin(x) - ----------------- + cos(x)
            x*sin(x) + cos(x)         
--------------------------------------
          x*sin(x) + cos(x)           
$$\frac{- \frac{x^{2} \cos^{2}{\left(x \right)}}{x \sin{\left(x \right)} + \cos{\left(x \right)}} - x \sin{\left(x \right)} + \cos{\left(x \right)}}{x \sin{\left(x \right)} + \cos{\left(x \right)}}$$
The third derivative [src]
                                 2                3    3             2              
                          3*x*cos (x)          2*x *cos (x)       3*x *cos(x)*sin(x)
-2*sin(x) - x*cos(x) - ----------------- + -------------------- + ------------------
                       x*sin(x) + cos(x)                      2   x*sin(x) + cos(x) 
                                           (x*sin(x) + cos(x))                      
------------------------------------------------------------------------------------
                                 x*sin(x) + cos(x)                                  
$$\frac{\frac{2 x^{3} \cos^{3}{\left(x \right)}}{\left(x \sin{\left(x \right)} + \cos{\left(x \right)}\right)^{2}} + \frac{3 x^{2} \sin{\left(x \right)} \cos{\left(x \right)}}{x \sin{\left(x \right)} + \cos{\left(x \right)}} - x \cos{\left(x \right)} - \frac{3 x \cos^{2}{\left(x \right)}}{x \sin{\left(x \right)} + \cos{\left(x \right)}} - 2 \sin{\left(x \right)}}{x \sin{\left(x \right)} + \cos{\left(x \right)}}$$
The graph
Derivative of log(x*sin(x)+cos(x))