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Derivative of sin(2pix/a)

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   /2*pi*x\
sin|------|
   \  a   /

$$\sin{\left(\frac{2 \pi x}{a} \right)}$$

sin(((2*pi)*x)/a)

Detail solution

Let $u = \frac{2 \pi x}{a}$ .
The derivative of sine is cosine:

$\frac{d}{d u} \sin{\left(u \right)} = \cos{\left(u \right)}$
Then, apply the chain rule. Multiply by $\frac{\partial}{\partial x} \frac{2 \pi x}{a}$ :
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: $2 \pi$
  So, the result is: $\frac{2 \pi}{a}$
The result of the chain rule is:

$\frac{2 \pi \cos{\left(\frac{2 \pi x}{a} \right)}}{a}$
Now simplify:

$\frac{2 \pi \cos{\left(\frac{2 \pi x}{a} \right)}}{a}$

The answer is:

$\frac{2 \pi \cos{\left(\frac{2 \pi x}{a} \right)}}{a}$

The first derivative [src]

        /2*pi*x\
2*pi*cos|------|
        \  a   /
----------------
       a

$$\frac{2 \pi \cos{\left(\frac{2 \pi x}{a} \right)}}{a}$$

Simplify

The second derivative [src]

     2    /2*pi*x\
-4*pi *sin|------|
          \  a   /
------------------
         2        
        a

$$- \frac{4 \pi^{2} \sin{\left(\frac{2 \pi x}{a} \right)}}{a^{2}}$$

Simplify

The third derivative [src]

     3    /2*pi*x\
-8*pi *cos|------|
          \  a   /
------------------
         3        
        a

$$- \frac{8 \pi^{3} \cos{\left(\frac{2 \pi x}{a} \right)}}{a^{3}}$$

Simplify

Derivative of sin(2*pi*x/a)