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Derivative of:
Derivative of x/16
Derivative of sin9x
Derivative of log(x+11)
Derivative of (ln(x+8)-3x+2)/x
Identical expressions
-4picos(pit/ six)/ three
minus 4 Pi co sinus of e of ( Pi t divide by 6) divide by 3
minus 4 Pi co sinus of e of ( Pi t divide by six) divide by three
-4picospit/6/3
-4picos(pit divide by 6) divide by 3
Similar expressions
4picos(pit/6)/3

The derivative of the function/ -4picos(pit/6)/3

Derivative of -4picos(pit/6)/3

The solution

You have entered [src]

         /pi*t\
-4*pi*cos|----|
         \ 6  /
---------------
       3

$$\frac{- 4 \pi \cos{\left(\frac{\pi t}{6} \right)}}{3}$$

((-4*pi)*cos((pi*t)/6))/3

Detail solution

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. Let $u = \frac{\pi t}{6}$ .
  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 $\frac{d}{d t} \frac{\pi t}{6}$ :
    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.
        
        Apply the power rule: $t$ goes to $1$
        
        So, the result is: $\pi$
      So, the result is: $\frac{\pi}{6}$
    The result of the chain rule is:
    
    $- \frac{\pi \sin{\left(\frac{\pi t}{6} \right)}}{6}$
  So, the result is: $\frac{2 \pi^{2} \sin{\left(\frac{\pi t}{6} \right)}}{3}$
So, the result is: $\frac{2 \pi^{2} \sin{\left(\frac{\pi t}{6} \right)}}{9}$
Now simplify:

$\frac{2 \pi^{2} \sin{\left(\frac{\pi t}{6} \right)}}{9}$

The answer is:

$\frac{2 \pi^{2} \sin{\left(\frac{\pi t}{6} \right)}}{9}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

    2    /pi*t\
2*pi *sin|----|
         \ 6  /
---------------
       9

$$\frac{2 \pi^{2} \sin{\left(\frac{\pi t}{6} \right)}}{9}$$

Simplify

The second derivative [src]

  3    /pi*t\
pi *cos|----|
       \ 6  /
-------------
      27

$$\frac{\pi^{3} \cos{\left(\frac{\pi t}{6} \right)}}{27}$$

Simplify

The third derivative [src]

   4    /pi*t\ 
-pi *sin|----| 
        \ 6  / 
---------------
      162

$$- \frac{\pi^{4} \sin{\left(\frac{\pi t}{6} \right)}}{162}$$

Simplify