Mister Exam

Lang:

Other calculators

How to use it?
Derivative of:
Derivative of (3*x-5)^6
Derivative of 6^(x^2-8x+28)
Derivative of y=2e^-x
Derivative of y=1/cos²x
Identical expressions
twelve *t*cos(t)/(t^ two + one)
12 multiply by t multiply by co sinus of e of (t) divide by (t squared plus 1)
twelve multiply by t multiply by co sinus of e of (t) divide by (t to the power of two plus one)
12*t*cos(t)/(t2+1)
12*t*cost/t2+1
12*t*cos(t)/(t²+1)
12*t*cos(t)/(t to the power of 2+1)
12tcos(t)/(t^2+1)
12tcos(t)/(t2+1)
12tcost/t2+1
12tcost/t^2+1
12*t*cos(t) divide by (t^2+1)
Similar expressions
12*t*cos(t)/(t^2-1)

The derivative of the function/ 12*t*cos(t)/(t^2+1)

Derivative of 12tcos(t)/(t^2+1)

The solution

You have entered [src]

12*t*cos(t)
-----------
    2      
   t  + 1

$$\frac{12 t \cos{\left(t \right)}}{t^{2} + 1}$$

((12*t)*cos(t))/(t^2 + 1)

Detail solution

Apply the quotient rule, which is:

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

$f{\left(t \right)} = 12 t \cos{\left(t \right)}$ and $g{\left(t \right)} = t^{2} + 1$ .

To find $\frac{d}{d t} f{\left(t \right)}$ :
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: $- 12 t \sin{\left(t \right)} + 12 \cos{\left(t \right)}$
To find $\frac{d}{d t} g{\left(t \right)}$ :
1. Differentiate $t^{2} + 1$ term by term:
  1. The derivative of the constant $1$ is zero.
  2. Apply the power rule: $t^{2}$ goes to $2 t$
  The result is: $2 t$
Now plug in to the quotient rule:

$\frac{- 24 t^{2} \cos{\left(t \right)} + \left(t^{2} + 1\right) \left(- 12 t \sin{\left(t \right)} + 12 \cos{\left(t \right)}\right)}{\left(t^{2} + 1\right)^{2}}$
Now simplify:

$- \frac{24 t^{2} \cos{\left(t \right)} + 12 \left(t^{2} + 1\right) \left(t \sin{\left(t \right)} - \cos{\left(t \right)}\right)}{\left(t^{2} + 1\right)^{2}}$

The answer is:

$- \frac{24 t^{2} \cos{\left(t \right)} + 12 \left(t^{2} + 1\right) \left(t \sin{\left(t \right)} - \cos{\left(t \right)}\right)}{\left(t^{2} + 1\right)^{2}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

                              2       
12*cos(t) - 12*t*sin(t)   24*t *cos(t)
----------------------- - ------------
          2                        2  
         t  + 1            / 2    \   
                           \t  + 1/

$$- \frac{24 t^{2} \cos{\left(t \right)}}{\left(t^{2} + 1\right)^{2}} + \frac{- 12 t \sin{\left(t \right)} + 12 \cos{\left(t \right)}}{t^{2} + 1}$$

Simplify

The second derivative [src]

   /                                                      /         2 \       \
   |                                                      |      4*t  |       |
   |                                                  2*t*|-1 + ------|*cos(t)|
   |                                                      |          2|       |
   |                       4*t*(-cos(t) + t*sin(t))       \     1 + t /       |
12*|-2*sin(t) - t*cos(t) + ------------------------ + ------------------------|
   |                                     2                          2         |
   \                                1 + t                      1 + t          /
-------------------------------------------------------------------------------
                                          2                                    
                                     1 + t

$$\frac{12 \left(- t \cos{\left(t \right)} + \frac{4 t \left(t \sin{\left(t \right)} - \cos{\left(t \right)}\right)}{t^{2} + 1} + \frac{2 t \left(\frac{4 t^{2}}{t^{2} + 1} - 1\right) \cos{\left(t \right)}}{t^{2} + 1} - 2 \sin{\left(t \right)}\right)}{t^{2} + 1}$$

Simplify

The third derivative [src]

   /                         /         2 \                                                          /         2 \       \
   |                         |      4*t  |                                                        2 |      2*t  |       |
   |                       6*|-1 + ------|*(-cos(t) + t*sin(t))                               24*t *|-1 + ------|*cos(t)|
   |                         |          2|                                                          |          2|       |
   |                         \     1 + t /                        6*t*(2*sin(t) + t*cos(t))         \     1 + t /       |
12*|-3*cos(t) + t*sin(t) - ------------------------------------ + ------------------------- - --------------------------|
   |                                           2                                 2                            2         |
   |                                      1 + t                             1 + t                     /     2\          |
   \                                                                                                  \1 + t /          /
-------------------------------------------------------------------------------------------------------------------------
                                                               2                                                         
                                                          1 + t

$$\frac{12 \left(- \frac{24 t^{2} \left(\frac{2 t^{2}}{t^{2} + 1} - 1\right) \cos{\left(t \right)}}{\left(t^{2} + 1\right)^{2}} + t \sin{\left(t \right)} + \frac{6 t \left(t \cos{\left(t \right)} + 2 \sin{\left(t \right)}\right)}{t^{2} + 1} - 3 \cos{\left(t \right)} - \frac{6 \left(t \sin{\left(t \right)} - \cos{\left(t \right)}\right) \left(\frac{4 t^{2}}{t^{2} + 1} - 1\right)}{t^{2} + 1}\right)}{t^{2} + 1}$$

Simplify

Other calculators

How to use it?

Identical expressions

Similar expressions

Derivative of 12tcos(t)/(t^2+1)

The graph:

Piecewise:

The solution

Other calculators

How to use it?

Identical expressions

Similar expressions

Derivative of 12*t*cos(t)/(t^2+1)

The graph:

Piecewise:

The solution

Derivative of 12tcos(t)/(t^2+1)