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(2*x-9)/(x-5)^2

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Derivative of:
Derivative of e^-1
Derivative of e^(2-x)
Derivative of e^x*cosx
Derivative of 6*x^3
Identical expressions
(two *x- nine)/(x- five)^ two
(2 multiply by x minus 9) divide by (x minus 5) squared
(two multiply by x minus nine) divide by (x minus five) to the power of two
(2*x-9)/(x-5)2
2*x-9/x-52
(2*x-9)/(x-5)²
(2*x-9)/(x-5) to the power of 2
(2x-9)/(x-5)^2
(2x-9)/(x-5)2
2x-9/x-52
2x-9/x-5^2
(2*x-9) divide by (x-5)^2
Similar expressions
(2*x+9)/(x-5)^2
(2*x-9)/(x+5)^2

The derivative of the function/ (2*x-9)/(x-5)^2

Derivative of (2*x-9)/(x-5)^2

The solution

You have entered [src]

2*x - 9 
--------
       2
(x - 5)

$$\frac{2 x - 9}{\left(x - 5\right)^{2}}$$

d /2*x - 9 \
--|--------|
dx|       2|
  \(x - 5) /

$$\frac{d}{d x} \frac{2 x - 9}{\left(x - 5\right)^{2}}$$

Transform

Detail solution

Apply the quotient rule, which is:

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

$f{\left(x \right)} = 2 x - 9$ and $g{\left(x \right)} = \left(x - 5\right)^{2}$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Differentiate $2 x - 9$ term by term:
  1. The derivative of the constant $-9$ 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. Let $u = x - 5$ .
2. Apply the power rule: $u^{2}$ goes to $2 u$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(x - 5\right)$ :
  1. Differentiate $x - 5$ term by term:
    1. The derivative of the constant $-5$ is zero.
    2. Apply the power rule: $x$ goes to $1$
    The result is: $1$
  The result of the chain rule is:
  
  $2 x - 10$
Now plug in to the quotient rule:

$\frac{2 \left(x - 5\right)^{2} - \left(2 x - 10\right) \left(2 x - 9\right)}{\left(x - 5\right)^{4}}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

   2       (10 - 2*x)*(2*x - 9)
-------- + --------------------
       2                4      
(x - 5)          (x - 5)

$$\frac{\left(- 2 x + 10\right) \left(2 x - 9\right)}{\left(x - 5\right)^{4}} + \frac{2}{\left(x - 5\right)^{2}}$$

Simplify

The second derivative [src]

  /     3*(-9 + 2*x)\
2*|-4 + ------------|
  \        -5 + x   /
---------------------
              3      
      (-5 + x)

$$\frac{2 \left(-4 + \frac{3 \cdot \left(2 x - 9\right)}{x - 5}\right)}{\left(x - 5\right)^{3}}$$

Simplify

The third derivative [src]

   /    2*(-9 + 2*x)\
12*|3 - ------------|
   \       -5 + x   /
---------------------
              4      
      (-5 + x)

$$\frac{12 \cdot \left(3 - \frac{2 \cdot \left(2 x - 9\right)}{x - 5}\right)}{\left(x - 5\right)^{4}}$$

Simplify

The graph

Derivative of (2*x-9)/(x-5)^2