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ln(1+sinx/cosx)
The derivative of the function/ ln(1+sinx/cosx)

Derivative of ln(1+sinx/cosx)

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

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The solution

You have entered [src] 
   /    sin(x)\
log|1 + ------|
   \    cos(x)/
log⁡(sin⁡(x)cos⁡(x)+1)\log{\left(\frac{\sin{\left(x \right)}}{\cos{\left(x \right)}} + 1 \right)} 
d /   /    sin(x)\\
--|log|1 + ------||
dx\   \    cos(x)//
ddxlog⁡(sin⁡(x)cos⁡(x)+1)\frac{d}{d x} \log{\left(\frac{\sin{\left(x \right)}}{\cos{\left(x \right)}} + 1 \right)}
Transform
Detail solution

  1. Let u=sin⁡(x)cos⁡(x)+1u = \frac{\sin{\left(x \right)}}{\cos{\left(x \right)}} + 1.

  2. The derivative of log⁡(u)\log{\left(u \right)} is 1u\frac{1}{u}.

  3. Then, apply the chain rule. Multiply by ddx(sin⁡(x)cos⁡(x)+1)\frac{d}{d x} \left(\frac{\sin{\left(x \right)}}{\cos{\left(x \right)}} + 1\right):

    1. Differentiate sin⁡(x)cos⁡(x)+1\frac{\sin{\left(x \right)}}{\cos{\left(x \right)}} + 1 term by term:

      1. The derivative of the constant 11 is zero.

      2. Apply the quotient rule, which is:

        ddxf(x)g(x)=−f(x)ddxg(x)+g(x)ddxf(x)g2(x)\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(x)=sin⁡(x)f{\left(x \right)} = \sin{\left(x \right)} and g(x)=cos⁡(x)g{\left(x \right)} = \cos{\left(x \right)}.

        To find ddxf(x)\frac{d}{d x} f{\left(x \right)}:

        1. The derivative of sine is cosine:

          ddxsin⁡(x)=cos⁡(x)\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}

        To find ddxg(x)\frac{d}{d x} g{\left(x \right)}:

        1. The derivative of cosine is negative sine:

          ddxcos⁡(x)=−sin⁡(x)\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}

        Now plug in to the quotient rule:

        sin⁡2(x)+cos⁡2(x)cos⁡2(x)\frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}}

      The result is: sin⁡2(x)+cos⁡2(x)cos⁡2(x)\frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}}

    The result of the chain rule is:

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

  4. Now simplify:

    222sin⁡(2x+π4)+2\frac{2 \sqrt{2}}{2 \sin{\left(2 x + \frac{\pi}{4} \right)} + \sqrt{2}}

The answer is:

222sin⁡(2x+π4)+2\frac{2 \sqrt{2}}{2 \sin{\left(2 x + \frac{\pi}{4} \right)} + \sqrt{2}}

The graph
02468-8-6-4-2-1010-100100
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
       2   
    sin (x)
1 + -------
       2   
    cos (x)
-----------
     sin(x)
 1 + ------
     cos(x)
sin⁡2(x)cos⁡2(x)+1sin⁡(x)cos⁡(x)+1\frac{\frac{\sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 1}{\frac{\sin{\left(x \right)}}{\cos{\left(x \right)}} + 1}
Simplify
The second derivative [src] 
              /         2              \
              |      sin (x)           |
              |  1 + -------           |
/       2   \ |         2              |
|    sin (x)| |      cos (x)   2*sin(x)|
|1 + -------|*|- ----------- + --------|
|       2   | |       sin(x)    cos(x) |
\    cos (x)/ |   1 + ------           |
              \       cos(x)           /
----------------------------------------
                   sin(x)               
               1 + ------               
                   cos(x)
(2sin⁡(x)cos⁡(x)−sin⁡2(x)cos⁡2(x)+1sin⁡(x)cos⁡(x)+1)(sin⁡2(x)cos⁡2(x)+1)sin⁡(x)cos⁡(x)+1\frac{\left(\frac{2 \sin{\left(x \right)}}{\cos{\left(x \right)}} - \frac{\frac{\sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 1}{\frac{\sin{\left(x \right)}}{\cos{\left(x \right)}} + 1}\right) \left(\frac{\sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 1\right)}{\frac{\sin{\left(x \right)}}{\cos{\left(x \right)}} + 1}
Simplify
The third derivative [src] 
  /                 3                                          2       \
  |    /       2   \                              /       2   \        |
  |    |    sin (x)|                              |    sin (x)|        |
  |    |1 + -------|                            3*|1 + -------| *sin(x)|
  |    |       2   |         4           2        |       2   |        |
  |    \    cos (x)/    3*sin (x)   4*sin (x)     \    cos (x)/        |
2*|1 + -------------- + --------- + --------- - -----------------------|
  |                2        4           2         /    sin(x)\         |
  |    /    sin(x)\      cos (x)     cos (x)      |1 + ------|*cos(x)  |
  |    |1 + ------|                               \    cos(x)/         |
  \    \    cos(x)/                                                    /
------------------------------------------------------------------------
                                   sin(x)                               
                               1 + ------                               
                                   cos(x)
2⋅(−3(sin⁡2(x)cos⁡2(x)+1)2sin⁡(x)(sin⁡(x)cos⁡(x)+1)cos⁡(x)+(sin⁡2(x)cos⁡2(x)+1)3(sin⁡(x)cos⁡(x)+1)2+3sin⁡4(x)cos⁡4(x)+4sin⁡2(x)cos⁡2(x)+1)sin⁡(x)cos⁡(x)+1\frac{2 \cdot \left(- \frac{3 \left(\frac{\sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 1\right)^{2} \sin{\left(x \right)}}{\left(\frac{\sin{\left(x \right)}}{\cos{\left(x \right)}} + 1\right) \cos{\left(x \right)}} + \frac{\left(\frac{\sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 1\right)^{3}}{\left(\frac{\sin{\left(x \right)}}{\cos{\left(x \right)}} + 1\right)^{2}} + \frac{3 \sin^{4}{\left(x \right)}}{\cos^{4}{\left(x \right)}} + \frac{4 \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 1\right)}{\frac{\sin{\left(x \right)}}{\cos{\left(x \right)}} + 1}
Simplify
The graph
Derivative of ln(1+sinx/cosx)
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