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Derivative of:
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Derivative of f(x)=x²+3x-2
Derivative of f(x)=2x⁵-3x²
Derivative of f(x)=2x-3x^2
Identical expressions
sin^ two (pi*x/l)
sinus of squared ( Pi multiply by x divide by l)
sinus of to the power of two ( Pi multiply by x divide by l)
sin2(pi*x/l)
sin2pi*x/l
sin²(pi*x/l)
sin to the power of 2(pi*x/l)
sin^2(pix/l)
sin2(pix/l)
sin2pix/l
sin^2pix/l
sin^2(pi*x divide by l)

The derivative of the function/ sin^2(pi*x/l)

Derivative of sin^2(pi*x/l)

The solution

You have entered [src]

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

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

sin((pi*x)/l)^2

Detail solution

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

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

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

The answer is:

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

The first derivative [src]

        /pi*x\    /pi*x\
2*pi*cos|----|*sin|----|
        \ l  /    \ l  /
------------------------
           l

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

Simplify

The second derivative [src]

    2 /   2/pi*x\      2/pi*x\\
2*pi *|cos |----| - sin |----||
      \    \ l  /       \ l  //
-------------------------------
                2              
               l

$$\frac{2 \pi^{2} \left(- \sin^{2}{\left(\frac{\pi x}{l} \right)} + \cos^{2}{\left(\frac{\pi x}{l} \right)}\right)}{l^{2}}$$

Simplify

The third derivative [src]

     3    /pi*x\    /pi*x\
-8*pi *cos|----|*sin|----|
          \ l  /    \ l  /
--------------------------
             3            
            l

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

Simplify

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Derivative of sin^2(pi*x/l)

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