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The derivative of the function/ pisin(2pit/3)

Derivative of pisin(2pit/3)

The solution

You have entered [src]

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

$$\pi \sin{\left(\frac{2 \pi t}{3} \right)}$$

pi*sin(((2*pi)*t)/3)

Detail solution

The derivative of a constant times a function is the constant times the derivative of the function.
1. Let $u = \frac{2 \pi t}{3}$ .
2. The derivative of sine is cosine:
  
  $\frac{d}{d u} \sin{\left(u \right)} = \cos{\left(u \right)}$
3. Then, apply the chain rule. Multiply by $\frac{d}{d t} \frac{2 \pi t}{3}$ :
  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: $2 \pi$
    So, the result is: $\frac{2 \pi}{3}$
  The result of the chain rule is:
  
  $\frac{2 \pi \cos{\left(\frac{2 \pi t}{3} \right)}}{3}$
So, the result is: $\frac{2 \pi^{2} \cos{\left(\frac{2 \pi t}{3} \right)}}{3}$
Now simplify:

$\frac{2 \pi^{2} \cos{\left(\frac{2 \pi t}{3} \right)}}{3}$

The answer is:

$\frac{2 \pi^{2} \cos{\left(\frac{2 \pi t}{3} \right)}}{3}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

    2    /2*pi*t\
2*pi *cos|------|
         \  3   /
-----------------
        3

$$\frac{2 \pi^{2} \cos{\left(\frac{2 \pi t}{3} \right)}}{3}$$

Simplify

The second derivative [src]

     3    /2*pi*t\
-4*pi *sin|------|
          \  3   /
------------------
        9

$$- \frac{4 \pi^{3} \sin{\left(\frac{2 \pi t}{3} \right)}}{9}$$

Simplify

The third derivative [src]

     4    /2*pi*t\
-8*pi *cos|------|
          \  3   /
------------------
        27

$$- \frac{8 \pi^{4} \cos{\left(\frac{2 \pi t}{3} \right)}}{27}$$

Simplify