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z*cos(z/(z+1))
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Similar expressions
z*cos(z/(z-1))

The derivative of the function/ z*cos(z/(z+1))

Derivative of z*cos(z/(z+1))

The solution

You have entered [src]

     /  z  \
z*cos|-----|
     \z + 1/

$$z \cos{\left(\frac{z}{z + 1} \right)}$$

z*cos(z/(z + 1))

Detail solution

Apply the product rule:

$\frac{d}{d z} f{\left(z \right)} g{\left(z \right)} = f{\left(z \right)} \frac{d}{d z} g{\left(z \right)} + g{\left(z \right)} \frac{d}{d z} f{\left(z \right)}$

$f{\left(z \right)} = z$ ; to find $\frac{d}{d z} f{\left(z \right)}$ :
1. Apply the power rule: $z$ goes to $1$
$g{\left(z \right)} = \cos{\left(\frac{z}{z + 1} \right)}$ ; to find $\frac{d}{d z} g{\left(z \right)}$ :
1. Let $u = \frac{z}{z + 1}$ .
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 $\frac{d}{d z} \frac{z}{z + 1}$ :
  1. Apply the quotient rule, which is:
    
    $\frac{d}{d z} \frac{f{\left(z \right)}}{g{\left(z \right)}} = \frac{- f{\left(z \right)} \frac{d}{d z} g{\left(z \right)} + g{\left(z \right)} \frac{d}{d z} f{\left(z \right)}}{g^{2}{\left(z \right)}}$
    
    $f{\left(z \right)} = z$ and $g{\left(z \right)} = z + 1$ .
    
    To find $\frac{d}{d z} f{\left(z \right)}$ :
    1. Apply the power rule: $z$ goes to $1$
    To find $\frac{d}{d z} g{\left(z \right)}$ :
    1. Differentiate $z + 1$ term by term:
      1. The derivative of the constant $1$ is zero.
      2. Apply the power rule: $z$ goes to $1$
      The result is: $1$
    Now plug in to the quotient rule:
    
    $\frac{1}{\left(z + 1\right)^{2}}$
  The result of the chain rule is:
  
  $- \frac{\sin{\left(\frac{z}{z + 1} \right)}}{\left(z + 1\right)^{2}}$
The result is: $- \frac{z \sin{\left(\frac{z}{z + 1} \right)}}{\left(z + 1\right)^{2}} + \cos{\left(\frac{z}{z + 1} \right)}$
Now simplify:

$- \frac{z \sin{\left(\frac{z}{z + 1} \right)}}{\left(z + 1\right)^{2}} + \cos{\left(\frac{z}{z + 1} \right)}$

The answer is:

$- \frac{z \sin{\left(\frac{z}{z + 1} \right)}}{\left(z + 1\right)^{2}} + \cos{\left(\frac{z}{z + 1} \right)}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

    /  1        z    \    /  z  \      /  z  \
- z*|----- - --------|*sin|-----| + cos|-----|
    |z + 1          2|    \z + 1/      \z + 1/
    \        (z + 1) /

$$- z \left(- \frac{z}{\left(z + 1\right)^{2}} + \frac{1}{z + 1}\right) \sin{\left(\frac{z}{z + 1} \right)} + \cos{\left(\frac{z}{z + 1} \right)}$$

Simplify

The second derivative [src]

             /                 /     /  z  \   /       z  \    /  z  \\\
             |               z*|2*sin|-----| + |-1 + -----|*cos|-----|||
/       z  \ |     /  z  \     \     \1 + z/   \     1 + z/    \1 + z//|
|-1 + -----|*|2*sin|-----| - ------------------------------------------|
\     1 + z/ \     \1 + z/                     1 + z                   /
------------------------------------------------------------------------
                                 1 + z

$$\frac{\left(\frac{z}{z + 1} - 1\right) \left(- \frac{z \left(\left(\frac{z}{z + 1} - 1\right) \cos{\left(\frac{z}{z + 1} \right)} + 2 \sin{\left(\frac{z}{z + 1} \right)}\right)}{z + 1} + 2 \sin{\left(\frac{z}{z + 1} \right)}\right)}{z + 1}$$

Simplify

The third derivative [src]

             /                                               /                           2                                       \\
             |                                               |     /  z  \   /       z  \     /  z  \     /       z  \    /  z  \||
             |                                             z*|6*sin|-----| - |-1 + -----| *sin|-----| + 6*|-1 + -----|*cos|-----|||
/       z  \ |       /  z  \     /       z  \    /  z  \     \     \1 + z/   \     1 + z/     \1 + z/     \     1 + z/    \1 + z//|
|-1 + -----|*|- 6*sin|-----| - 3*|-1 + -----|*cos|-----| + -----------------------------------------------------------------------|
\     1 + z/ \       \1 + z/     \     1 + z/    \1 + z/                                    1 + z                                 /
-----------------------------------------------------------------------------------------------------------------------------------
                                                                     2                                                             
                                                              (1 + z)

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

Simplify

The graph

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Derivative of z*cos(z/(z+1))

The graph:

Piecewise:

The solution