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Derivative of:
Derivative of e^-1
Derivative of 2*x^5
Derivative of sin(2*x^2+3)
Derivative of f(x)=5
Identical expressions
(-x- three)/((x- one)^ three)
( minus x minus 3) divide by ((x minus 1) cubed )
( minus x minus three) divide by ((x minus one) to the power of three)
(-x-3)/((x-1)3)
-x-3/x-13
(-x-3)/((x-1)³)
(-x-3)/((x-1) to the power of 3)
-x-3/x-1^3
(-x-3) divide by ((x-1)^3)
Similar expressions
(-x-3)/((x+1)^3)
(x-3)/((x-1)^3)
(-x+3)/((x-1)^3)

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

Derivative of (-x-3)/((x-1)^3)

The solution

You have entered [src]

 -x - 3 
--------
       3
(x - 1)

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

(-x - 3)/(x - 1)^3

Detail solution

Apply the quotient rule, which is:

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

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

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

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

   /     5*(3 + x)\
12*|-3 + ---------|
   \       -1 + x /
-------------------
             5     
     (-1 + x)

$$\frac{12 \left(-3 + \frac{5 \left(x + 3\right)}{x - 1}\right)}{\left(x - 1\right)^{5}}$$

Simplify