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sin^2x/cosx
The derivative of the function/ sin^2x/cosx

Derivative of sin^2x/cosx

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

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The solution

You have entered [src] 
   2   
sin (x)
-------
 cos(x)
sin⁡2(x)cos⁡(x)\frac{\sin^{2}{\left(x \right)}}{\cos{\left(x \right)}} 
  /   2   \
d |sin (x)|
--|-------|
dx\ cos(x)/
ddxsin⁡2(x)cos⁡(x)\frac{d}{d x} \frac{\sin^{2}{\left(x \right)}}{\cos{\left(x \right)}}
Transform
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⁡2(x)f{\left(x \right)} = \sin^{2}{\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. Let u=sin⁡(x)u = \sin{\left(x \right)}.

    2. Apply the power rule: u2u^{2} goes to 2u2 u

    3. Then, apply the chain rule. Multiply by ddxsin⁡(x)\frac{d}{d x} \sin{\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)}

      The result of the chain rule is:

      2sin⁡(x)cos⁡(x)2 \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⁡3(x)+2sin⁡(x)cos⁡2(x)cos⁡2(x)\frac{\sin^{3}{\left(x \right)} + 2 \sin{\left(x \right)} \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}}

  2. Now simplify:

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

The answer is:

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

The graph
02468-8-6-4-2-1010-20002000
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
              3   
           sin (x)
2*sin(x) + -------
              2   
           cos (x)
sin⁡3(x)cos⁡2(x)+2sin⁡(x)\frac{\sin^{3}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 2 \sin{\left(x \right)}
Simplify
The second derivative [src] 
                                /         2   \
     2           2         2    |    2*sin (x)|
2*cos (x) + 2*sin (x) + sin (x)*|1 + ---------|
                                |        2    |
                                \     cos (x) /
-----------------------------------------------
                     cos(x)
(2sin⁡2(x)cos⁡2(x)+1)sin⁡2(x)+2sin⁡2(x)+2cos⁡2(x)cos⁡(x)\frac{\left(\frac{2 \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 1\right) \sin^{2}{\left(x \right)} + 2 \sin^{2}{\left(x \right)} + 2 \cos^{2}{\left(x \right)}}{\cos{\left(x \right)}}
Simplify
The third derivative [src] 
/                                                  /         2   \\       
|                                             2    |    6*sin (x)||       
|                                          sin (x)*|5 + ---------||       
|       /   2         2   \         2              |        2    ||       
|     6*\sin (x) - cos (x)/   12*sin (x)           \     cos (x) /|       
|-2 - --------------------- + ---------- + -----------------------|*sin(x)
|               2                 2                   2           |       
\            cos (x)           cos (x)             cos (x)        /
((6sin⁡2(x)cos⁡2(x)+5)sin⁡2(x)cos⁡2(x)+12sin⁡2(x)cos⁡2(x)−2−6(sin⁡2(x)−cos⁡2(x))cos⁡2(x))sin⁡(x)\left(\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{12 \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} - 2 - \frac{6 \left(\sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)}\right)}{\cos^{2}{\left(x \right)}}\right) \sin{\left(x \right)}
Simplify
The graph
Derivative of sin^2x/cosx
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