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The derivative of the function/ 1/(2x-5)

Derivative of 1/(2x-5)

The solution

You have entered [src]

     1   
1*-------
  2*x - 5

$$1 \cdot \frac{1}{2 x - 5}$$

d /     1   \
--|1*-------|
dx\  2*x - 5/

$$\frac{d}{d x} 1 \cdot \frac{1}{2 x - 5}$$

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 $g{\left(x \right)} = 2 x - 5$ .

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. Differentiate $2 x - 5$ term by term:
  1. The derivative of the constant $-5$ is zero.
  2. 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: $2$
  The result is: $2$
Now plug in to the quotient rule:

$- \frac{2}{\left(2 x - 5\right)^{2}}$

The answer is:

$- \frac{2}{\left(2 x - 5\right)^{2}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

   -2     
----------
         2
(2*x - 5)

$$- \frac{2}{\left(2 x - 5\right)^{2}}$$

Simplify

The second derivative [src]

     8     
-----------
          3
(-5 + 2*x)

$$\frac{8}{\left(2 x - 5\right)^{3}}$$

Simplify

The third derivative [src]

    -48    
-----------
          4
(-5 + 2*x)

$$- \frac{48}{\left(2 x - 5\right)^{4}}$$

Simplify

The graph

Derivative of 1/(2x-5)