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sinx/(1+tanx)
The derivative of the function/ sinx/(1+tanx)

Derivative of sinx/(1+tanx)

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

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The solution

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

  1. 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)=tan⁡(x)+1g{\left(x \right)} = \tan{\left(x \right)} + 1.

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

      1. The derivative of the constant 11 is zero.

      2. Rewrite the function to be differentiated:

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

      3. 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)}}

    Now plug in to the quotient rule:

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

  2. Now simplify:

    −2sin⁡(x+π4)tan⁡2(x)+1cos⁡(x)(tan⁡(x)+1)2\frac{- \sqrt{2} \sin{\left(x + \frac{\pi}{4} \right)} \tan^{2}{\left(x \right)} + \frac{1}{\cos{\left(x \right)}}}{\left(\tan{\left(x \right)} + 1\right)^{2}}

The answer is:

−2sin⁡(x+π4)tan⁡2(x)+1cos⁡(x)(tan⁡(x)+1)2\frac{- \sqrt{2} \sin{\left(x + \frac{\pi}{4} \right)} \tan^{2}{\left(x \right)} + \frac{1}{\cos{\left(x \right)}}}{\left(\tan{\left(x \right)} + 1\right)^{2}}

The graph
02468-8-6-4-2-1010-25002500
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
             /        2   \       
  cos(x)     \-1 - tan (x)/*sin(x)
---------- + ---------------------
1 + tan(x)                   2    
                 (1 + tan(x))
cos⁡(x)tan⁡(x)+1+(−tan⁡2(x)−1)sin⁡(x)(tan⁡(x)+1)2\frac{\cos{\left(x \right)}}{\tan{\left(x \right)} + 1} + \frac{\left(- \tan^{2}{\left(x \right)} - 1\right) \sin{\left(x \right)}}{\left(\tan{\left(x \right)} + 1\right)^{2}}
Simplify
The second derivative [src] 
 /                                         /         2            \                \ 
 |                           /       2   \ |  1 + tan (x)         |                | 
 |  /       2   \          2*\1 + tan (x)/*|- ----------- + tan(x)|*sin(x)         | 
 |2*\1 + tan (x)/*cos(x)                   \   1 + tan(x)         /                | 
-|---------------------- + ----------------------------------------------- + sin(x)| 
 \      1 + tan(x)                            1 + tan(x)                           / 
-------------------------------------------------------------------------------------
                                      1 + tan(x)
−2(tan⁡(x)−tan⁡2(x)+1tan⁡(x)+1)(tan⁡2(x)+1)sin⁡(x)tan⁡(x)+1+sin⁡(x)+2(tan⁡2(x)+1)cos⁡(x)tan⁡(x)+1tan⁡(x)+1- \frac{\frac{2 \left(\tan{\left(x \right)} - \frac{\tan^{2}{\left(x \right)} + 1}{\tan{\left(x \right)} + 1}\right) \left(\tan^{2}{\left(x \right)} + 1\right) \sin{\left(x \right)}}{\tan{\left(x \right)} + 1} + \sin{\left(x \right)} + \frac{2 \left(\tan^{2}{\left(x \right)} + 1\right) \cos{\left(x \right)}}{\tan{\left(x \right)} + 1}}{\tan{\left(x \right)} + 1}
Simplify
The third derivative [src] 
                                                                                                     /                               2                         \       
                                                                                                     |                  /       2   \      /       2   \       |       
                                                   /         2            \            /       2   \ |         2      3*\1 + tan (x)/    6*\1 + tan (x)/*tan(x)|       
                                     /       2   \ |  1 + tan (x)         |          2*\1 + tan (x)/*|1 + 3*tan (x) + ---------------- - ----------------------|*sin(x)
            /       2   \          6*\1 + tan (x)/*|- ----------- + tan(x)|*cos(x)                   |                             2           1 + tan(x)      |       
          3*\1 + tan (x)/*sin(x)                   \   1 + tan(x)         /                          \                 (1 + tan(x))                            /       
-cos(x) + ---------------------- - ----------------------------------------------- - ----------------------------------------------------------------------------------
                1 + tan(x)                            1 + tan(x)                                                         1 + tan(x)                                    
-----------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                               1 + tan(x)
−6(tan⁡(x)−tan⁡2(x)+1tan⁡(x)+1)(tan⁡2(x)+1)cos⁡(x)tan⁡(x)+1−2(tan⁡2(x)+1)(3tan⁡2(x)−6(tan⁡2(x)+1)tan⁡(x)tan⁡(x)+1+1+3(tan⁡2(x)+1)2(tan⁡(x)+1)2)sin⁡(x)tan⁡(x)+1−cos⁡(x)+3(tan⁡2(x)+1)sin⁡(x)tan⁡(x)+1tan⁡(x)+1\frac{- \frac{6 \left(\tan{\left(x \right)} - \frac{\tan^{2}{\left(x \right)} + 1}{\tan{\left(x \right)} + 1}\right) \left(\tan^{2}{\left(x \right)} + 1\right) \cos{\left(x \right)}}{\tan{\left(x \right)} + 1} - \frac{2 \left(\tan^{2}{\left(x \right)} + 1\right) \left(3 \tan^{2}{\left(x \right)} - \frac{6 \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)}}{\tan{\left(x \right)} + 1} + 1 + \frac{3 \left(\tan^{2}{\left(x \right)} + 1\right)^{2}}{\left(\tan{\left(x \right)} + 1\right)^{2}}\right) \sin{\left(x \right)}}{\tan{\left(x \right)} + 1} - \cos{\left(x \right)} + \frac{3 \left(\tan^{2}{\left(x \right)} + 1\right) \sin{\left(x \right)}}{\tan{\left(x \right)} + 1}}{\tan{\left(x \right)} + 1}
Simplify
The graph
Derivative of sinx/(1+tanx)
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