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sin(x)/cos(x)^2

Derivative of sin(x)/cos(x)^2

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

The graph:

from to

Piecewise:

The solution

You have entered [src]
 sin(x)
-------
   2   
cos (x)
$$\frac{\sin{\left(x \right)}}{\cos^{2}{\left(x \right)}}$$
sin(x)/cos(x)^2
Detail solution
  1. Apply the quotient rule, which is:

    and .

    To find :

    1. The derivative of sine is cosine:

    To find :

    1. Let .

    2. Apply the power rule: goes to

    3. Then, apply the chain rule. Multiply by :

      1. The derivative of cosine is negative sine:

      The result of the chain rule is:

    Now plug in to the quotient rule:

  2. Now simplify:


The answer is:

The graph
The first derivative [src]
               2   
 cos(x)   2*sin (x)
------- + ---------
   2          3    
cos (x)    cos (x) 
$$\frac{2 \sin^{2}{\left(x \right)}}{\cos^{3}{\left(x \right)}} + \frac{\cos{\left(x \right)}}{\cos^{2}{\left(x \right)}}$$
The second derivative [src]
/         2   \       
|    6*sin (x)|       
|5 + ---------|*sin(x)
|        2    |       
\     cos (x) /       
----------------------
          2           
       cos (x)        
$$\frac{\left(\frac{6 \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 5\right) \sin{\left(x \right)}}{\cos^{2}{\left(x \right)}}$$
The third derivative [src]
                           /         2   \
                      2    |    3*sin (x)|
                 8*sin (x)*|2 + ---------|
          2                |        2    |
    12*sin (x)             \     cos (x) /
5 + ---------- + -------------------------
        2                    2            
     cos (x)              cos (x)         
------------------------------------------
                  cos(x)                  
$$\frac{\frac{8 \left(\frac{3 \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 2\right) \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + \frac{12 \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 5}{\cos{\left(x \right)}}$$
The graph
Derivative of sin(x)/cos(x)^2