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Derivative of (3*x-2)^7
Identical expressions
ln^ two *cos(x/ four)
ln squared multiply by co sinus of e of (x divide by 4)
ln to the power of two multiply by co sinus of e of (x divide by four)
ln2*cos(x/4)
ln2*cosx/4
ln²*cos(x/4)
ln to the power of 2*cos(x/4)
ln^2cos(x/4)
ln2cos(x/4)
ln2cosx/4
ln^2cosx/4
ln^2*cos(x divide by 4)

The derivative of the function/ ln^2*cos(x/4)

Derivative of ln^2*cos(x/4)

The solution

You have entered [src]

   2       /x\
log (x)*cos|-|
           \4/

$$\log{\left(x \right)}^{2} \cos{\left(\frac{x}{4} \right)}$$

log(x)^2*cos(x/4)

Detail solution

Apply the product rule:

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

$f{\left(x \right)} = \log{\left(x \right)}^{2}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
1. Let $u = \log{\left(x \right)}$ .
2. Apply the power rule: $u^{2}$ goes to $2 u$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \log{\left(x \right)}$ :
  1. The derivative of $\log{\left(x \right)}$ is $\frac{1}{x}$ .
  The result of the chain rule is:
  
  $\frac{2 \log{\left(x \right)}}{x}$
$g{\left(x \right)} = \cos{\left(\frac{x}{4} \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
1. Let $u = \frac{x}{4}$ .
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 $\frac{d}{d x} \frac{x}{4}$ :
  1. 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: $\frac{1}{4}$
  The result of the chain rule is:
  
  $- \frac{\sin{\left(\frac{x}{4} \right)}}{4}$
The result is: $- \frac{\log{\left(x \right)}^{2} \sin{\left(\frac{x}{4} \right)}}{4} + \frac{2 \log{\left(x \right)} \cos{\left(\frac{x}{4} \right)}}{x}$
Now simplify:

$\frac{\left(- x \log{\left(x \right)} \sin{\left(\frac{x}{4} \right)} + 8 \cos{\left(\frac{x}{4} \right)}\right) \log{\left(x \right)}}{4 x}$

The answer is:

$\frac{\left(- x \log{\left(x \right)} \sin{\left(\frac{x}{4} \right)} + 8 \cos{\left(\frac{x}{4} \right)}\right) \log{\left(x \right)}}{4 x}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

     2       /x\        /x\       
  log (x)*sin|-|   2*cos|-|*log(x)
             \4/        \4/       
- -------------- + ---------------
        4                 x

$$- \frac{\log{\left(x \right)}^{2} \sin{\left(\frac{x}{4} \right)}}{4} + \frac{2 \log{\left(x \right)} \cos{\left(\frac{x}{4} \right)}}{x}$$

Simplify

The second derivative [src]

 /   2       /x\             /x\                      /x\\
 |log (x)*cos|-|   log(x)*sin|-|   2*(-1 + log(x))*cos|-||
 |           \4/             \4/                      \4/|
-|-------------- + ------------- + ----------------------|
 |      16               x                    2          |
 \                                           x           /

$$- (\frac{\log{\left(x \right)}^{2} \cos{\left(\frac{x}{4} \right)}}{16} + \frac{\log{\left(x \right)} \sin{\left(\frac{x}{4} \right)}}{x} + \frac{2 \left(\log{\left(x \right)} - 1\right) \cos{\left(\frac{x}{4} \right)}}{x^{2}})$$

Simplify

The third derivative [src]

   2       /x\                        /x\        /x\                             /x\
log (x)*sin|-|   2*(-3 + 2*log(x))*cos|-|   3*cos|-|*log(x)   3*(-1 + log(x))*sin|-|
           \4/                        \4/        \4/                             \4/
-------------- + ------------------------ - --------------- + ----------------------
      64                     3                    8*x                     2         
                            x                                          2*x

$$\frac{\log{\left(x \right)}^{2} \sin{\left(\frac{x}{4} \right)}}{64} - \frac{3 \log{\left(x \right)} \cos{\left(\frac{x}{4} \right)}}{8 x} + \frac{3 \left(\log{\left(x \right)} - 1\right) \sin{\left(\frac{x}{4} \right)}}{2 x^{2}} + \frac{2 \left(2 \log{\left(x \right)} - 3\right) \cos{\left(\frac{x}{4} \right)}}{x^{3}}$$

Simplify

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Identical expressions

Derivative of ln^2*cos(x/4)

The graph:

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The solution