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Derivative of:
Derivative of atan(sqrt(x))
Derivative of 3*x^5
Derivative of 3*x^2*e^x
Derivative of 3*cos(3*x)
Identical expressions
(three *x- one)/(two *x+ five)
(3 multiply by x minus 1) divide by (2 multiply by x plus 5)
(three multiply by x minus one) divide by (two multiply by x plus five)
(3x-1)/(2x+5)
3x-1/2x+5
(3*x-1) divide by (2*x+5)
Similar expressions
(3*x-1)/(2*x-5)
3x-1/2x+5
y=3x-1/2x+5
(3*x+1)/(2*x+5)

The derivative of the function/ (3*x-1)/(2*x+5)

Derivative of (3x-1)/(2x+5)

The solution

You have entered [src]

3*x - 1
-------
2*x + 5

$$\frac{3 x - 1}{2 x + 5}$$

(3*x - 1)/(2*x + 5)

Detail solution

Apply the quotient rule, which is:

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

$f{\left(x \right)} = 3 x - 1$ and $g{\left(x \right)} = 2 x + 5$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Differentiate $3 x - 1$ term by term:
  1. The derivative of the constant $-1$ 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: $3$
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:

$\frac{17}{\left(2 x + 5\right)^{2}}$

The answer is:

$\frac{17}{\left(2 x + 5\right)^{2}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

   3      2*(3*x - 1)
------- - -----------
2*x + 5             2
           (2*x + 5)

$$\frac{3}{2 x + 5} - \frac{2 \left(3 x - 1\right)}{\left(2 x + 5\right)^{2}}$$

Simplify

The second derivative [src]

  /     2*(-1 + 3*x)\
4*|-3 + ------------|
  \       5 + 2*x   /
---------------------
               2     
      (5 + 2*x)

$$\frac{4 \left(-3 + \frac{2 \left(3 x - 1\right)}{2 x + 5}\right)}{\left(2 x + 5\right)^{2}}$$

Simplify

The third derivative [src]

   /    2*(-1 + 3*x)\
24*|3 - ------------|
   \      5 + 2*x   /
---------------------
               3     
      (5 + 2*x)

$$\frac{24 \left(3 - \frac{2 \left(3 x - 1\right)}{2 x + 5}\right)}{\left(2 x + 5\right)^{3}}$$

Simplify