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y(x)=1/x(x-2)

The derivative of the function/ y(x)=1/x(x+2)

Derivative of y(x)=1/x(x+2)

The solution

You have entered [src]

x + 2
-----
  x

$$\frac{x + 2}{x}$$

(x + 2)/x

Detail solution

Apply the quotient rule, which is:

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

$f{\left(x \right)} = x + 2$ and $g{\left(x \right)} = x$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
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: $1$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Apply the power rule: $x$ goes to $1$
Now plug in to the quotient rule:

$- \frac{2}{x^{2}}$

The answer is:

$- \frac{2}{x^{2}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

1   x + 2
- - -----
x      2 
      x

$$\frac{1}{x} - \frac{x + 2}{x^{2}}$$

Simplify

The second derivative [src]

  /     2 + x\
2*|-1 + -----|
  \       x  /
--------------
       2      
      x

$$\frac{2 \left(-1 + \frac{x + 2}{x}\right)}{x^{2}}$$

Simplify

The third derivative [src]

  /    2 + x\
6*|1 - -----|
  \      x  /
-------------
       3     
      x

$$\frac{6 \left(1 - \frac{x + 2}{x}\right)}{x^{3}}$$

Simplify

The graph

Derivative of y(x)=1/x(x+2)