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Derivative of:
Derivative of 1
Derivative of x*cos(x)
Derivative of 6x
Derivative of (x^2+25)/x
Identical expressions
a/r^ twelve -b/r^ six
a divide by r to the power of 12 minus b divide by r to the power of 6
a divide by r to the power of twelve minus b divide by r to the power of six
a/r12-b/r6
a/r^12-b/r⁶
a divide by r^12-b divide by r^6
Similar expressions
a/r^12+b/r^6

The derivative of the function/ a/r^12-b/r^6

Derivative of a/r^12-b/r^6

The solution

You have entered [src]

 a    b 
--- - --
 12    6
r     r

$$\frac{a}{r^{12}} - \frac{b}{r^{6}}$$

a/r^12 - b/r^6

Detail solution

Differentiate $\frac{a}{r^{12}} - \frac{b}{r^{6}}$ term by term:
1. The derivative of a constant times a function is the constant times the derivative of the function.
  1. Let $u = r^{12}$ .
  2. Apply the power rule: $\frac{1}{u}$ goes to $- \frac{1}{u^{2}}$
  3. Then, apply the chain rule. Multiply by $\frac{d}{d r} r^{12}$ :
    1. Apply the power rule: $r^{12}$ goes to $12 r^{11}$
    The result of the chain rule is:
    
    $- \frac{12}{r^{13}}$
  So, the result is: $- \frac{12 a}{r^{13}}$
2. The derivative of a constant times a function is the constant times the derivative of the function.
  1. Let $u = r^{6}$ .
  2. Apply the power rule: $\frac{1}{u}$ goes to $- \frac{1}{u^{2}}$
  3. Then, apply the chain rule. Multiply by $\frac{d}{d r} r^{6}$ :
    1. Apply the power rule: $r^{6}$ goes to $6 r^{5}$
    The result of the chain rule is:
    
    $- \frac{6}{r^{7}}$
  So, the result is: $\frac{6 b}{r^{7}}$
The result is: $- \frac{12 a}{r^{13}} + \frac{6 b}{r^{7}}$
Now simplify:

$\frac{6 \left(- 2 a + b r^{6}\right)}{r^{13}}$

The answer is:

$\frac{6 \left(- 2 a + b r^{6}\right)}{r^{13}}$

The first derivative [src]

  12*a   6*b
- ---- + ---
   13      7
  r       r

$$- \frac{12 a}{r^{13}} + \frac{6 b}{r^{7}}$$

Simplify

The second derivative [src]

  /       26*a\
6*|-7*b + ----|
  |         6 |
  \        r  /
---------------
        8      
       r

$$\frac{6 \left(\frac{26 a}{r^{6}} - 7 b\right)}{r^{8}}$$

Simplify

The third derivative [src]

    /      13*a\
168*|2*b - ----|
    |        6 |
    \       r  /
----------------
        9       
       r

$$\frac{168 \left(- \frac{13 a}{r^{6}} + 2 b\right)}{r^{9}}$$

Simplify