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Identical expressions
cos((pi*x)/ two , eight)^ two
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cos((pi*x)/2,8)2
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cos((pi*x) divide by 2,8)^2

The derivative of the function/ cos((pi*x)/2,8)^2

Derivative of cos((pi*x)/2,8)^2

The solution

You have entered [src]

   2/pi*x\
cos |----|
    \14/5/

$$\cos^{2}{\left(\frac{\pi x}{\frac{14}{5}} \right)}$$

cos((pi*x)/(14/5))^2

Detail solution

Let $u = \cos{\left(\frac{\pi x}{\frac{14}{5}} \right)}$ .
Apply the power rule: $u^{2}$ goes to $2 u$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \cos{\left(\frac{\pi x}{\frac{14}{5}} \right)}$ :
1. Let $u = \frac{\pi x}{\frac{14}{5}}$ .
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 $\frac{d}{d x} \frac{\pi x}{\frac{14}{5}}$ :
  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: $\frac{5 \pi}{14}$
  The result of the chain rule is:
  
  $- \frac{5 \pi \sin{\left(\frac{\pi x}{\frac{14}{5}} \right)}}{14}$
The result of the chain rule is:

$- \frac{5 \pi \sin{\left(\frac{\pi x}{\frac{14}{5}} \right)} \cos{\left(\frac{\pi x}{\frac{14}{5}} \right)}}{7}$
Now simplify:

$- \frac{5 \pi \sin{\left(\frac{5 \pi x}{7} \right)}}{14}$

The answer is:

$- \frac{5 \pi \sin{\left(\frac{5 \pi x}{7} \right)}}{14}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

         /pi*x\    /pi*x\
-5*pi*cos|----|*sin|----|
         \14/5/    \14/5/
-------------------------
            7

$$- \frac{5 \pi \sin{\left(\frac{\pi x}{\frac{14}{5}} \right)} \cos{\left(\frac{\pi x}{\frac{14}{5}} \right)}}{7}$$

Simplify

The second derivative [src]

     2 /   2/5*pi*x\      2/5*pi*x\\
25*pi *|sin |------| - cos |------||
       \    \  14  /       \  14  //
------------------------------------
                 98

$$\frac{25 \pi^{2} \left(\sin^{2}{\left(\frac{5 \pi x}{14} \right)} - \cos^{2}{\left(\frac{5 \pi x}{14} \right)}\right)}{98}$$

Simplify

The third derivative [src]

      3    /5*pi*x\    /5*pi*x\
125*pi *cos|------|*sin|------|
           \  14  /    \  14  /
-------------------------------
              343

$$\frac{125 \pi^{3} \sin{\left(\frac{5 \pi x}{14} \right)} \cos{\left(\frac{5 \pi x}{14} \right)}}{343}$$

Simplify

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Derivative of cos((pi*x)/2,8)^2

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The solution