Derivative of u*cos(u\v)

You have entered [src]

     /u\
u*cos|-|
     \v/

$$u \cos{\left(\frac{u}{v} \right)}$$

u*cos(u/v)

Detail solution

The derivative of a constant times a function is the constant times the derivative of the function.
1. Let $u = \frac{u}{v}$ .
2. The derivative of cosine is negative sine:
  
  $\frac{d}{d u} \cos{\left(u \right)} = - \sin{\left(u \right)}$
3. Then, apply the chain rule. Multiply by $\frac{\partial}{\partial v} \frac{u}{v}$ :
  1. The derivative of a constant times a function is the constant times the derivative of the function.
    1. Apply the power rule: $\frac{1}{v}$ goes to $- \frac{1}{v^{2}}$
    So, the result is: $- \frac{u}{v^{2}}$
  The result of the chain rule is:
  
  $\frac{u \sin{\left(\frac{u}{v} \right)}}{v^{2}}$
So, the result is: $\frac{u^{2} \sin{\left(\frac{u}{v} \right)}}{v^{2}}$

The answer is:

$\frac{u^{2} \sin{\left(\frac{u}{v} \right)}}{v^{2}}$

The first derivative [src]

 2    /u\
u *sin|-|
      \v/
---------
     2   
    v

$$\frac{u^{2} \sin{\left(\frac{u}{v} \right)}}{v^{2}}$$

Simplify

The second derivative [src]

    /                /u\\ 
    |           u*cos|-|| 
  2 |     /u\        \v/| 
-u *|2*sin|-| + --------| 
    \     \v/      v    / 
--------------------------
             3            
            v

$$- \frac{u^{2} \left(\frac{u \cos{\left(\frac{u}{v} \right)}}{v} + 2 \sin{\left(\frac{u}{v} \right)}\right)}{v^{3}}$$

Simplify

The third derivative [src]

   /            2    /u\          /u\\
   |           u *sin|-|   6*u*cos|-||
 2 |     /u\         \v/          \v/|
u *|6*sin|-| - --------- + ----------|
   |     \v/        2          v     |
   \               v                 /
--------------------------------------
                   4                  
                  v

$$\frac{u^{2} \left(- \frac{u^{2} \sin{\left(\frac{u}{v} \right)}}{v^{2}} + \frac{6 u \cos{\left(\frac{u}{v} \right)}}{v} + 6 \sin{\left(\frac{u}{v} \right)}\right)}{v^{4}}$$

Simplify

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Derivative of u*cos(u\v)

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