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Derivative of:
Derivative of (-6)/x^4
Derivative of -1/y
Derivative of y=ln(1/x)
Derivative of y=log6sin4x
Identical expressions
40pi*cos((pit)/ six)
40 Pi multiply by co sinus of e of (( Pi t) divide by 6)
40 Pi multiply by co sinus of e of (( Pi t) divide by six)
40picos((pit)/6)
40picospit/6
40pi*cos((pit) divide by 6)

The derivative of the function/ 40pi*cos((pit)/6)

Derivative of 40pi*cos((pit)/6)

The solution

You have entered [src]

         /pi*t\
40*pi*cos|----|
         \ 6  /

$$40 \pi \cos{\left(\frac{\pi t}{6} \right)}$$

(40*pi)*cos((pi*t)/6)

Detail solution

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.
      1. 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: $- \frac{20 \pi^{2} \sin{\left(\frac{\pi t}{6} \right)}}{3}$
Now simplify:

$- \frac{20 \pi^{2} \sin{\left(\frac{\pi t}{6} \right)}}{3}$

The answer is:

$- \frac{20 \pi^{2} \sin{\left(\frac{\pi t}{6} \right)}}{3}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

      2    /pi*t\
-20*pi *sin|----|
           \ 6  /
-----------------
        3

$$- \frac{20 \pi^{2} \sin{\left(\frac{\pi t}{6} \right)}}{3}$$

Simplify

The second derivative [src]

      3    /pi*t\
-10*pi *cos|----|
           \ 6  /
-----------------
        9

$$- \frac{10 \pi^{3} \cos{\left(\frac{\pi t}{6} \right)}}{9}$$

Simplify

The third derivative [src]

    4    /pi*t\
5*pi *sin|----|
         \ 6  /
---------------
       27

$$\frac{5 \pi^{4} \sin{\left(\frac{\pi t}{6} \right)}}{27}$$

Simplify