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Derivative of (sin(x))^2
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Derivative of sin(x)+cos(x)
Derivative of asin(x)
Identical expressions
(x^ two - five)/(2x+ one)
(x squared minus 5) divide by (2x plus 1)
(x to the power of two minus five) divide by (2x plus one)
(x2-5)/(2x+1)
x2-5/2x+1
(x²-5)/(2x+1)
(x to the power of 2-5)/(2x+1)
x^2-5/2x+1
(x^2-5) divide by (2x+1)
Similar expressions
(x^2-5)/(2x-1)
(x^2+5)/(2x+1)

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

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

The solution

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

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

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

Detail solution

Apply the quotient rule, which is:

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

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

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Differentiate $x^{2} - 5$ term by term:
  1. The derivative of the constant $-5$ is zero.
  2. Apply the power rule: $x^{2}$ goes to $2 x$
  The result is: $2 x$
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:

$\frac{- 2 x^{2} + 2 x \left(2 x + 1\right) + 10}{\left(2 x + 1\right)^{2}}$
Now simplify:

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

The answer is:

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

The graph

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Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

   /       /      2\          \
   |     4*\-5 + x /     4*x  |
12*|-1 - ----------- + -------|
   |               2   1 + 2*x|
   \      (1 + 2*x)           /
-------------------------------
                    2          
           (1 + 2*x)

$$\frac{12 \left(\frac{4 x}{2 x + 1} - 1 - \frac{4 \left(x^{2} - 5\right)}{\left(2 x + 1\right)^{2}}\right)}{\left(2 x + 1\right)^{2}}$$

Simplify

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

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The solution