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Derivative of:
Derivative of f(x)=x^2
Derivative of (x+1)*(x+2)^2*(x+3)^3
Derivative of tan(x)-x^2
Derivative of tan(2*x)/(1-cot(2*x))
Identical expressions
(three *x+ one)/(x- two)^ two
(3 multiply by x plus 1) divide by (x minus 2) squared
(three multiply by x plus one) divide by (x minus two) to the power of two
(3*x+1)/(x-2)2
3*x+1/x-22
(3*x+1)/(x-2)²
(3*x+1)/(x-2) to the power of 2
(3x+1)/(x-2)^2
(3x+1)/(x-2)2
3x+1/x-22
3x+1/x-2^2
(3*x+1) divide by (x-2)^2
Similar expressions
(3*x-1)/(x-2)^2
(3*x+1)/(x+2)^2

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

Derivative of (3*x+1)/(x-2)^2

The solution

You have entered [src]

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

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

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

Detail solution

Apply the quotient rule, which is:

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

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

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. Let $u = x - 2$ .
2. Apply the power rule: $u^{2}$ goes to $2 u$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(x - 2\right)$ :
  1. Differentiate $x - 2$ term by term:
    1. The derivative of the constant $-2$ is zero.
    2. Apply the power rule: $x$ goes to $1$
    The result is: $1$
  The result of the chain rule is:
  
  $2 x - 4$
Now plug in to the quotient rule:

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

$- \frac{3 x + 8}{\left(x - 2\right)^{3}}$

The answer is:

$- \frac{3 x + 8}{\left(x - 2\right)^{3}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

   3       (4 - 2*x)*(3*x + 1)
-------- + -------------------
       2                4     
(x - 2)          (x - 2)

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

Simplify

The second derivative [src]

  /     1 + 3*x\
6*|-2 + -------|
  \      -2 + x/
----------------
           3    
   (-2 + x)

$$\frac{6 \left(-2 + \frac{3 x + 1}{x - 2}\right)}{\left(x - 2\right)^{3}}$$

Simplify

3-я производная [src]

  /    4*(1 + 3*x)\
6*|9 - -----------|
  \       -2 + x  /
-------------------
             4     
     (-2 + x)

$$\frac{6 \left(9 - \frac{4 \left(3 x + 1\right)}{x - 2}\right)}{\left(x - 2\right)^{4}}$$

Simplify

The third derivative [src]

  /    4*(1 + 3*x)\
6*|9 - -----------|
  \       -2 + x  /
-------------------
             4     
     (-2 + x)

$$\frac{6 \left(9 - \frac{4 \left(3 x + 1\right)}{x - 2}\right)}{\left(x - 2\right)^{4}}$$

Simplify