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Derivative of:
Derivative of e^2*x
Derivative of x^sin(x)
Derivative of sqrt(x^2+1)
Derivative of 4/x^2
Identical expressions
(- six)/(x^ two + six *x)
( minus 6) divide by (x squared plus 6 multiply by x)
( minus six) divide by (x to the power of two plus six multiply by x)
(-6)/(x2+6*x)
-6/x2+6*x
(-6)/(x²+6*x)
(-6)/(x to the power of 2+6*x)
(-6)/(x^2+6x)
(-6)/(x2+6x)
-6/x2+6x
-6/x^2+6x
(-6) divide by (x^2+6*x)
Similar expressions
(6)/(x^2+6*x)
(-6)/(x^2-6*x)

The derivative of the function/ (-6)/(x^2+6*x)

Derivative of (-6)/(x^2+6*x)

The solution

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  -6    
--------
 2      
x  + 6*x

$$- \frac{6}{x^{2} + 6 x}$$

-6/(x^2 + 6*x)

Detail solution

The derivative of a constant times a function is the constant times the derivative of the function.
1. Let $u = x^{2} + 6 x$ .
2. Apply the power rule: $\frac{1}{u}$ goes to $- \frac{1}{u^{2}}$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(x^{2} + 6 x\right)$ :
  1. Differentiate $x^{2} + 6 x$ term by term:
    1. Apply the power rule: $x^{2}$ goes to $2 x$
    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: $6$
    The result is: $2 x + 6$
  The result of the chain rule is:
  
  $- \frac{2 x + 6}{\left(x^{2} + 6 x\right)^{2}}$
So, the result is: $\frac{6 \left(2 x + 6\right)}{\left(x^{2} + 6 x\right)^{2}}$
Now simplify:

$\frac{12 \left(x + 3\right)}{x^{2} \left(x + 6\right)^{2}}$

The answer is:

$\frac{12 \left(x + 3\right)}{x^{2} \left(x + 6\right)^{2}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

-6*(-6 - 2*x)
-------------
           2 
 / 2      \  
 \x  + 6*x/

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

Simplify

The second derivative [src]

   /             2\
   |    4*(3 + x) |
12*|1 - ----------|
   \    x*(6 + x) /
-------------------
     2        2    
    x *(6 + x)

$$\frac{12 \left(1 - \frac{4 \left(x + 3\right)^{2}}{x \left(x + 6\right)}\right)}{x^{2} \left(x + 6\right)^{2}}$$

Simplify

The third derivative [src]

     /             2\        
     |    2*(3 + x) |        
-144*|1 - ----------|*(3 + x)
     \    x*(6 + x) /        
-----------------------------
          3        3         
         x *(6 + x)

$$- \frac{144 \left(1 - \frac{2 \left(x + 3\right)^{2}}{x \left(x + 6\right)}\right) \left(x + 3\right)}{x^{3} \left(x + 6\right)^{3}}$$

Simplify

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Derivative of (-6)/(x^2+6*x)

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The solution