Mister Exam

Lang:

Other calculators

How to use it?
Derivative of:
Derivative of x^2*sin(2*x-3)
Derivative of (x+1)^2*(x-5)-6
Derivative of log(11*x)
Derivative of log(t-1)
Identical expressions
sin(x)/(one +log(sin(x)))
sinus of (x) divide by (1 plus logarithm of ( sinus of (x)))
sinus of (x) divide by (one plus logarithm of ( sinus of (x)))
sinx/1+logsinx
sin(x) divide by (1+log(sin(x)))
Similar expressions
sin(x)/(1-log(sin(x)))
sinx/(1+log(sinx))

The derivative of the function/ sin(x)/(1+log(sin(x)))

Derivative of sin(x)/(1+log(sin(x)))

The solution

You have entered [src]

     sin(x)    
---------------
1 + log(sin(x))

$$\frac{\sin{\left(x \right)}}{\log{\left(\sin{\left(x \right)} \right)} + 1}$$

sin(x)/(1 + log(sin(x)))

Detail solution

Apply the quotient rule, which is:

$\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$

$f{\left(x \right)} = \sin{\left(x \right)}$ and $g{\left(x \right)} = \log{\left(\sin{\left(x \right)} \right)} + 1$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. The derivative of sine is cosine:
  
  $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Differentiate $\log{\left(\sin{\left(x \right)} \right)} + 1$ term by term:
  1. The derivative of the constant $1$ is zero.
  2. Let $u = \sin{\left(x \right)}$ .
  3. The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
  4. Then, apply the chain rule. Multiply by $\frac{d}{d x} \sin{\left(x \right)}$ :
    1. The derivative of sine is cosine:
      
      $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
    The result of the chain rule is:
    
    $\frac{\cos{\left(x \right)}}{\sin{\left(x \right)}}$
  The result is: $\frac{\cos{\left(x \right)}}{\sin{\left(x \right)}}$
Now plug in to the quotient rule:

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

     cos(x)             cos(x)      
--------------- - ------------------
1 + log(sin(x))                    2
                  (1 + log(sin(x)))

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

The graph

Other calculators

How to use it?

Identical expressions

Similar expressions

Derivative of sin(x)/(1+log(sin(x)))

The graph:

Piecewise:

The solution