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sinx/cosx
The derivative of the function/ sinx/cosx

Derivative of sinx/cosx

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

The graph:

from to

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The solution

You have entered [src] 
sin(x)
------
cos(x)
sin⁡(x)cos⁡(x)\frac{\sin{\left(x \right)}}{\cos{\left(x \right)}} 
sin(x)/cos(x)
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)=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)}}

  2. Now simplify:

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

The answer is:

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

The graph
02468-8-6-4-2-1010-10001000
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
       2   
    sin (x)
1 + -------
       2   
    cos (x)
sin⁡2(x)cos⁡2(x)+1\frac{\sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 1
Simplify
The second derivative [src] 
/         2   \       
|    2*sin (x)|       
|2 + ---------|*sin(x)
|        2    |       
\     cos (x) /       
----------------------
        cos(x)
(2sin⁡2(x)cos⁡2(x)+2)sin⁡(x)cos⁡(x)\frac{\left(\frac{2 \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 2\right) \sin{\left(x \right)}}{\cos{\left(x \right)}}
Simplify
The third derivative [src] 
                        /         2   \
                   2    |    6*sin (x)|
                sin (x)*|5 + ---------|
         2              |        2    |
    3*sin (x)           \     cos (x) /
2 + --------- + -----------------------
        2                  2           
     cos (x)            cos (x)
(6sin⁡2(x)cos⁡2(x)+5)sin⁡2(x)cos⁡2(x)+3sin⁡2(x)cos⁡2(x)+2\frac{\left(\frac{6 \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 5\right) \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + \frac{3 \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 2
Simplify
The graph
Derivative of sinx/cosx
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