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Derivative of:
Derivative of e^2*x
Derivative of cos(x)^2
Derivative of x^sin(x)
Derivative of sin(x^3)
Identical expressions
y=((4x- five)/(2x+ one))
y equally ((4x minus 5) divide by (2x plus 1))
y equally ((4x minus five) divide by (2x plus one))
y=4x-5/2x+1
y=((4x-5) divide by (2x+1))
Similar expressions
y=((4x+5)/(2x+1))
y=((4x-5)/(2x-1))

The derivative of the function/ y=((4x-5)/(2x+1))

Derivative of y=((4x-5)/(2x+1))

The solution

You have entered [src]

4*x - 5
-------
2*x + 1

$$\frac{4 x - 5}{2 x + 1}$$

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

Detail solution

Apply the quotient rule, which is:

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

$f{\left(x \right)} = 4 x - 5$ and $g{\left(x \right)} = 2 x + 1$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Differentiate $4 x - 5$ 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: $4$
  The result is: $4$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Differentiate $2 x + 1$ 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: $2$
  The result is: $2$
Now plug in to the quotient rule:

$\frac{14}{\left(2 x + 1\right)^{2}}$

The answer is:

$\frac{14}{\left(2 x + 1\right)^{2}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

   4      2*(4*x - 5)
------- - -----------
2*x + 1             2
           (2*x + 1)

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

Simplify

The second derivative [src]

  /     -5 + 4*x\
8*|-2 + --------|
  \     1 + 2*x /
-----------------
             2   
    (1 + 2*x)

$$\frac{8 \left(-2 + \frac{4 x - 5}{2 x + 1}\right)}{\left(2 x + 1\right)^{2}}$$

Simplify

The third derivative [src]

   /    -5 + 4*x\
48*|2 - --------|
   \    1 + 2*x /
-----------------
             3   
    (1 + 2*x)

$$\frac{48 \left(2 - \frac{4 x - 5}{2 x + 1}\right)}{\left(2 x + 1\right)^{3}}$$

Simplify