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Derivative of:
Derivative of sin(x)/cos(x)
Derivative of cos(x)^(3)
Derivative of tan(x^3)
Derivative of 1/(2*sqrt(x))
Identical expressions
3sin(pi*x/ six)- one
3 sinus of ( Pi multiply by x divide by 6) minus 1
3 sinus of ( Pi multiply by x divide by six) minus one
3sin(pix/6)-1
3sinpix/6-1
3sin(pi*x divide by 6)-1
Similar expressions
3sin(pi*x/6)+1

The derivative of the function/ 3sin(pi*x/6)-1

Derivative of 3sin(pi*x/6)-1

The solution

You have entered [src]

     /pi*x\    
3*sin|----| - 1
     \ 6  /

$$3 \sin{\left(\frac{\pi x}{6} \right)} - 1$$

3*sin((pi*x)/6) - 1

Detail solution

Differentiate $3 \sin{\left(\frac{\pi x}{6} \right)} - 1$ term by term:
1. The derivative of a constant times a function is the constant times the derivative of the function.
  1. Let $u = \frac{\pi x}{6}$ .
  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 $\frac{d}{d x} \frac{\pi x}{6}$ :
    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: $x$ goes to $1$
        
        So, the result is: $\pi$
      So, the result is: $\frac{\pi}{6}$
    The result of the chain rule is:
    
    $\frac{\pi \cos{\left(\frac{\pi x}{6} \right)}}{6}$
  So, the result is: $\frac{\pi \cos{\left(\frac{\pi x}{6} \right)}}{2}$
2. The derivative of the constant $-1$ is zero.
The result is: $\frac{\pi \cos{\left(\frac{\pi x}{6} \right)}}{2}$
Now simplify:

$\frac{\pi \cos{\left(\frac{\pi x}{6} \right)}}{2}$

The answer is:

$\frac{\pi \cos{\left(\frac{\pi x}{6} \right)}}{2}$

The graph

Plot the graph f(x)

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The first derivative [src]

      /pi*x\
pi*cos|----|
      \ 6  /
------------
     2

$$\frac{\pi \cos{\left(\frac{\pi x}{6} \right)}}{2}$$

Simplify

The second derivative [src]

   2    /pi*x\ 
-pi *sin|----| 
        \ 6  / 
---------------
       12

$$- \frac{\pi^{2} \sin{\left(\frac{\pi x}{6} \right)}}{12}$$

Simplify

The third derivative [src]

   3    /pi*x\ 
-pi *cos|----| 
        \ 6  / 
---------------
       72

$$- \frac{\pi^{3} \cos{\left(\frac{\pi x}{6} \right)}}{72}$$

Simplify