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The derivative of the function/ (2x+5)/(3x-2)

Derivative of (2x+5)/(3x-2)

The solution

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2*x + 5
-------
3*x - 2

$$\frac{2 x + 5}{3 x - 2}$$

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

Detail solution

Apply the quotient rule, which is:

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

$f{\left(x \right)} = 2 x + 5$ and $g{\left(x \right)} = 3 x - 2$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
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: $2$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Differentiate $3 x - 2$ term by term:
  1. The derivative of the constant $-2$ 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: $3$
  The result is: $3$
Now plug in to the quotient rule:

$- \frac{19}{\left(3 x - 2\right)^{2}}$

The answer is:

$- \frac{19}{\left(3 x - 2\right)^{2}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

   /    3*(5 + 2*x)\
54*|2 - -----------|
   \      -2 + 3*x /
--------------------
              3     
    (-2 + 3*x)

$$\frac{54 \left(- \frac{3 \left(2 x + 5\right)}{3 x - 2} + 2\right)}{\left(3 x - 2\right)^{3}}$$

Simplify

The graph

Derivative of (2x+5)/(3x-2)