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cos(pi*x/ two)*ln(one -x)
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The derivative of the function/ cos(pi*x/2)*ln(1-x)

Derivative of cos(pix/2)ln(1-x)

The solution

You have entered [src]

   /pi*x\           
cos|----|*log(1 - x)
   \ 2  /

$$\log{\left(1 - x \right)} \cos{\left(\frac{\pi x}{2} \right)}$$

cos((pi*x)/2)*log(1 - x)

Detail solution

Apply the product rule:

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

$f{\left(x \right)} = \cos{\left(\frac{\pi x}{2} \right)}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
1. Let $u = \frac{\pi x}{2}$ .
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 $\frac{d}{d x} \frac{\pi x}{2}$ :
  1. The derivative of a constant times a function is the constant times the derivative of the function.
    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: $\pi$
    So, the result is: $\frac{\pi}{2}$
  The result of the chain rule is:
  
  $- \frac{\pi \sin{\left(\frac{\pi x}{2} \right)}}{2}$
$g{\left(x \right)} = \log{\left(1 - x \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
1. Let $u = 1 - x$ .
2. The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(1 - x\right)$ :
  1. Differentiate $1 - x$ term by term:
    1. The derivative of the constant $1$ is zero.
    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: $-1$
    The result is: $-1$
  The result of the chain rule is:
  
  $- \frac{1}{1 - x}$
The result is: $- \frac{\pi \log{\left(1 - x \right)} \sin{\left(\frac{\pi x}{2} \right)}}{2} - \frac{\cos{\left(\frac{\pi x}{2} \right)}}{1 - x}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

     /pi*x\                    /pi*x\
  cos|----|   pi*log(1 - x)*sin|----|
     \ 2  /                    \ 2  /
- --------- - -----------------------
    1 - x                2

$$- \frac{\pi \log{\left(1 - x \right)} \sin{\left(\frac{\pi x}{2} \right)}}{2} - \frac{\cos{\left(\frac{\pi x}{2} \right)}}{1 - x}$$

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

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Derivative of cos(pix/2)ln(1-x)

The graph:

Piecewise:

The solution

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Derivative of cos(pi*x/2)*ln(1-x)

The graph:

Piecewise:

The solution

Derivative of cos(pix/2)ln(1-x)