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Derivative of:
Derivative of x^3/cosx
Derivative of (x-6)x^3
Derivative of y=x(x-1)
Derivative of y=5+12x-x³
Identical expressions
((two x)/(x+2))-3sinx
((2x) divide by (x plus 2)) minus 3 sinus of x
((two x) divide by (x plus 2)) minus 3 sinus of x
2x/x+2-3sinx
((2x) divide by (x+2))-3sinx
Similar expressions
((2x)/(x-2))-3sinx
((2x)/(x+2))+3sinx

The derivative of the function/ ((2x)/(x+2))-3sinx

Derivative of ((2x)/(x+2))-3sinx

The solution

You have entered [src]

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

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

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

Detail solution

Differentiate $\frac{2 x}{x + 2} - 3 \sin{\left(x \right)}$ term by term:
1. Apply the quotient rule, which is:
  
  $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
  
  $f{\left(x \right)} = 2 x$ and $g{\left(x \right)} = x + 2$ .
  
  To find $\frac{d}{d x} f{\left(x \right)}$ :
  1. The derivative of a constant times a function is the constant times the derivative of the function.
    1. Apply the power rule: $x$ goes to $1$
    So, the result is: $2$
  To find $\frac{d}{d x} g{\left(x \right)}$ :
  1. Differentiate $x + 2$ term by term:
    1. The derivative of the constant $2$ is zero.
    2. Apply the power rule: $x$ goes to $1$
    The result is: $1$
  Now plug in to the quotient rule:
  
  $\frac{4}{\left(x + 2\right)^{2}}$
2. The derivative of a constant times a function is the constant times the derivative of the function.
  1. The derivative of sine is cosine:
    
    $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
  So, the result is: $- 3 \cos{\left(x \right)}$
The result is: $- 3 \cos{\left(x \right)} + \frac{4}{\left(x + 2\right)^{2}}$

The answer is:

$- 3 \cos{\left(x \right)} + \frac{4}{\left(x + 2\right)^{2}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

     4                    4*x   
- -------- + 3*sin(x) + --------
         2                     3
  (2 + x)               (2 + x)

$$\frac{4 x}{\left(x + 2\right)^{3}} + 3 \sin{\left(x \right)} - \frac{4}{\left(x + 2\right)^{2}}$$

Simplify

The third derivative [src]

  /   4         4*x            \
3*|-------- - -------- + cos(x)|
  |       3          4         |
  \(2 + x)    (2 + x)          /

$$3 \left(- \frac{4 x}{\left(x + 2\right)^{4}} + \cos{\left(x \right)} + \frac{4}{\left(x + 2\right)^{3}}\right)$$

Simplify