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The derivative of the function/ 3*sin(x)/cos(x)

Derivative of 3*sin(x)/cos(x)

The solution

You have entered [src]

3*sin(x)
--------
 cos(x)

$$\frac{3 \sin{\left(x \right)}}{\cos{\left(x \right)}}$$

d /3*sin(x)\
--|--------|
dx\ cos(x) /

$$\frac{d}{d x} \frac{3 \sin{\left(x \right)}}{\cos{\left(x \right)}}$$

Transform

Detail solution

The derivative of a constant times a function is the constant times the derivative of the function.
1. Apply the quotient rule, which is:
  
  $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
  
  $f{\left(x \right)} = \sin{\left(x \right)}$ and $g{\left(x \right)} = \cos{\left(x \right)}$ .
  
  To find $\frac{d}{d x} f{\left(x \right)}$ :
  1. The derivative of sine is cosine:
    
    $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
  To find $\frac{d}{d x} g{\left(x \right)}$ :
  1. The derivative of cosine is negative sine:
    
    $\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}$
  Now plug in to the quotient rule:
  
  $\frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}}$
So, the result is: $\frac{3 \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right)}{\cos^{2}{\left(x \right)}}$
Now simplify:

$\frac{3}{\cos^{2}{\left(x \right)}}$

The answer is:

$\frac{3}{\cos^{2}{\left(x \right)}}$

The graph

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The first derivative [src]

         2   
    3*sin (x)
3 + ---------
        2    
     cos (x)

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

  /                        /         2   \\
  |                   2    |    6*sin (x)||
  |                sin (x)*|5 + ---------||
  |         2              |        2    ||
  |    3*sin (x)           \     cos (x) /|
3*|2 + --------- + -----------------------|
  |        2                  2           |
  \     cos (x)            cos (x)        /

$$3 \left(\frac{\left(\frac{6 \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 5\right) \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + \frac{3 \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 2\right)$$

Simplify

The graph

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Derivative of 3*sin(x)/cos(x)

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The solution