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The derivative of the function/ 1/(2x)

Derivative of 1/(2x)

The solution

You have entered [src]

   1 
1*---
  2*x

$$1 \cdot \frac{1}{2 x}$$

d /   1 \
--|1*---|
dx\  2*x/

$$\frac{d}{d x} 1 \cdot \frac{1}{2 x}$$

Transform

Detail solution

Apply the quotient rule, which is:

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

$f{\left(x \right)} = 1$ and $g{\left(x \right)} = 2 x$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. The derivative of the constant $1$ is zero.
To find $\frac{d}{d x} g{\left(x \right)}$ :
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: $2$
Now plug in to the quotient rule:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

   1  
- --- 
  2*x 
------
  x

$$- \frac{\frac{1}{2} \cdot \frac{1}{x}}{x}$$

Simplify

The second derivative [src]

1 
--
 3
x

$$\frac{1}{x^{3}}$$

Simplify

The third derivative [src]

-3 
---
  4
 x

$$- \frac{3}{x^{4}}$$

Simplify

The graph

Derivative of 1/(2x)