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The derivative of the function/ (sin(t)-t*cos(t))/(cos(t)+t*sin(t))

Derivative of (sin(t)-tcos(t))/(cos(t)+tsin(t))

The solution

You have entered [src]

sin(t) - t*cos(t)
-----------------
cos(t) + t*sin(t)

$$\frac{- t \cos{\left(t \right)} + \sin{\left(t \right)}}{t \sin{\left(t \right)} + \cos{\left(t \right)}}$$

(sin(t) - t*cos(t))/(cos(t) + t*sin(t))

Detail solution

Apply the quotient rule, which is:

$\frac{d}{d t} \frac{f{\left(t \right)}}{g{\left(t \right)}} = \frac{- f{\left(t \right)} \frac{d}{d t} g{\left(t \right)} + g{\left(t \right)} \frac{d}{d t} f{\left(t \right)}}{g^{2}{\left(t \right)}}$

$f{\left(t \right)} = - t \cos{\left(t \right)} + \sin{\left(t \right)}$ and $g{\left(t \right)} = t \sin{\left(t \right)} + \cos{\left(t \right)}$ .

To find $\frac{d}{d t} f{\left(t \right)}$ :
1. Differentiate $- t \cos{\left(t \right)} + \sin{\left(t \right)}$ term by term:
  1. The derivative of a constant times a function is the constant times the derivative of the function.
    1. Apply the product rule:
      
      $\frac{d}{d t} f{\left(t \right)} g{\left(t \right)} = f{\left(t \right)} \frac{d}{d t} g{\left(t \right)} + g{\left(t \right)} \frac{d}{d t} f{\left(t \right)}$
      
      $f{\left(t \right)} = t$ ; to find $\frac{d}{d t} f{\left(t \right)}$ :
      1. Apply the power rule: $t$ goes to $1$
      $g{\left(t \right)} = \cos{\left(t \right)}$ ; to find $\frac{d}{d t} g{\left(t \right)}$ :
      1. The derivative of cosine is negative sine:
        
        $\frac{d}{d t} \cos{\left(t \right)} = - \sin{\left(t \right)}$
      The result is: $- t \sin{\left(t \right)} + \cos{\left(t \right)}$
    So, the result is: $t \sin{\left(t \right)} - \cos{\left(t \right)}$
  2. The derivative of sine is cosine:
    
    $\frac{d}{d t} \sin{\left(t \right)} = \cos{\left(t \right)}$
  The result is: $t \sin{\left(t \right)}$
To find $\frac{d}{d t} g{\left(t \right)}$ :
1. Differentiate $t \sin{\left(t \right)} + \cos{\left(t \right)}$ term by term:
  1. Apply the product rule:
    
    $\frac{d}{d t} f{\left(t \right)} g{\left(t \right)} = f{\left(t \right)} \frac{d}{d t} g{\left(t \right)} + g{\left(t \right)} \frac{d}{d t} f{\left(t \right)}$
    
    $f{\left(t \right)} = t$ ; to find $\frac{d}{d t} f{\left(t \right)}$ :
    1. Apply the power rule: $t$ goes to $1$
    $g{\left(t \right)} = \sin{\left(t \right)}$ ; to find $\frac{d}{d t} g{\left(t \right)}$ :
    1. The derivative of sine is cosine:
      
      $\frac{d}{d t} \sin{\left(t \right)} = \cos{\left(t \right)}$
    The result is: $t \cos{\left(t \right)} + \sin{\left(t \right)}$
  2. The derivative of cosine is negative sine:
    
    $\frac{d}{d t} \cos{\left(t \right)} = - \sin{\left(t \right)}$
  The result is: $t \cos{\left(t \right)}$
Now plug in to the quotient rule:

$\frac{t \left(t \sin{\left(t \right)} + \cos{\left(t \right)}\right) \sin{\left(t \right)} - t \left(- t \cos{\left(t \right)} + \sin{\left(t \right)}\right) \cos{\left(t \right)}}{\left(t \sin{\left(t \right)} + \cos{\left(t \right)}\right)^{2}}$
Now simplify:

$\frac{t^{2}}{\left(t \sin{\left(t \right)} + \cos{\left(t \right)}\right)^{2}}$

The answer is:

$\frac{t^{2}}{\left(t \sin{\left(t \right)} + \cos{\left(t \right)}\right)^{2}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

     t*sin(t)       t*(sin(t) - t*cos(t))*cos(t)
----------------- - ----------------------------
cos(t) + t*sin(t)                          2    
                        (cos(t) + t*sin(t))

$$\frac{t \sin{\left(t \right)}}{t \sin{\left(t \right)} + \cos{\left(t \right)}} - \frac{t \left(- t \cos{\left(t \right)} + \sin{\left(t \right)}\right) \cos{\left(t \right)}}{\left(t \sin{\left(t \right)} + \cos{\left(t \right)}\right)^{2}}$$

Simplify

The second derivative [src]

                                /                           2    2     \                              
                                |                        2*t *cos (t)  |                              
           (-sin(t) + t*cos(t))*|-cos(t) + t*sin(t) + -----------------|      2                       
                                \                     t*sin(t) + cos(t)/   2*t *cos(t)*sin(t)         
t*cos(t) - ------------------------------------------------------------- - ------------------ + sin(t)
                                 t*sin(t) + cos(t)                         t*sin(t) + cos(t)          
------------------------------------------------------------------------------------------------------
                                          t*sin(t) + cos(t)

$$\frac{- \frac{2 t^{2} \sin{\left(t \right)} \cos{\left(t \right)}}{t \sin{\left(t \right)} + \cos{\left(t \right)}} + t \cos{\left(t \right)} + \sin{\left(t \right)} - \frac{\left(t \cos{\left(t \right)} - \sin{\left(t \right)}\right) \left(\frac{2 t^{2} \cos^{2}{\left(t \right)}}{t \sin{\left(t \right)} + \cos{\left(t \right)}} + t \sin{\left(t \right)} - \cos{\left(t \right)}\right)}{t \sin{\left(t \right)} + \cos{\left(t \right)}}}{t \sin{\left(t \right)} + \cos{\left(t \right)}}$$

Simplify

The third derivative [src]

                                           /                             3    3                    2            2              \                                                                                       
                                           |                          6*t *cos (t)          6*t*cos (t)      6*t *cos(t)*sin(t)|                                        /                           2    2     \       
                      (-sin(t) + t*cos(t))*|2*sin(t) + t*cos(t) - -------------------- + ----------------- - ------------------|                                        |                        2*t *cos (t)  |       
                                           |                                         2   t*sin(t) + cos(t)   t*sin(t) + cos(t) |                                    3*t*|-cos(t) + t*sin(t) + -----------------|*sin(t)
                                           \                      (t*sin(t) + cos(t))                                          /   3*t*(t*cos(t) + sin(t))*cos(t)       \                     t*sin(t) + cos(t)/       
2*cos(t) - t*sin(t) - ---------------------------------------------------------------------------------------------------------- - ------------------------------ + ---------------------------------------------------
                                                                  t*sin(t) + cos(t)                                                      t*sin(t) + cos(t)                           t*sin(t) + cos(t)                 
-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                                   t*sin(t) + cos(t)

$$\frac{- t \sin{\left(t \right)} - \frac{3 t \left(t \cos{\left(t \right)} + \sin{\left(t \right)}\right) \cos{\left(t \right)}}{t \sin{\left(t \right)} + \cos{\left(t \right)}} + \frac{3 t \left(\frac{2 t^{2} \cos^{2}{\left(t \right)}}{t \sin{\left(t \right)} + \cos{\left(t \right)}} + t \sin{\left(t \right)} - \cos{\left(t \right)}\right) \sin{\left(t \right)}}{t \sin{\left(t \right)} + \cos{\left(t \right)}} + 2 \cos{\left(t \right)} - \frac{\left(t \cos{\left(t \right)} - \sin{\left(t \right)}\right) \left(- \frac{6 t^{3} \cos^{3}{\left(t \right)}}{\left(t \sin{\left(t \right)} + \cos{\left(t \right)}\right)^{2}} - \frac{6 t^{2} \sin{\left(t \right)} \cos{\left(t \right)}}{t \sin{\left(t \right)} + \cos{\left(t \right)}} + t \cos{\left(t \right)} + \frac{6 t \cos^{2}{\left(t \right)}}{t \sin{\left(t \right)} + \cos{\left(t \right)}} + 2 \sin{\left(t \right)}\right)}{t \sin{\left(t \right)} + \cos{\left(t \right)}}}{t \sin{\left(t \right)} + \cos{\left(t \right)}}$$

Simplify

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Derivative of (sin(t)-tcos(t))/(cos(t)+tsin(t))

The graph:

Piecewise:

The solution

Other calculators

How to use it?

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Derivative of (sin(t)-t*cos(t))/(cos(t)+t*sin(t))

The graph:

Piecewise:

The solution

Derivative of (sin(t)-tcos(t))/(cos(t)+tsin(t))