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Derivative of:
Derivative of x^3/cosx
Derivative of 3^-x
Derivative of (x-6)x^3
Derivative of (x-5)^2*e^x-7
Identical expressions
3pi/ two *sin(pit/ two)
3 Pi divide by 2 multiply by sinus of ( Pi t divide by 2)
3 Pi divide by two multiply by sinus of ( Pi t divide by two)
3pi/2sin(pit/2)
3pi/2sinpit/2
3pi divide by 2*sin(pit divide by 2)

The derivative of the function/ 3pi/2*sin(pit/2)

Derivative of 3pi/2*sin(pit/2)

The solution

You have entered [src]

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

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

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

Detail solution

The derivative of a constant times a function is the constant times the derivative of the function.
1. Let $u = \frac{\pi t}{2}$ .
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 $\frac{d}{d t} \frac{\pi t}{2}$ :
  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: $\frac{\pi}{2}$
  The result of the chain rule is:
  
  $\frac{\pi \cos{\left(\frac{\pi t}{2} \right)}}{2}$
So, the result is: $\frac{3 \pi^{2} \cos{\left(\frac{\pi t}{2} \right)}}{4}$
Now simplify:

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

The answer is:

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

The graph

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Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

     3    /pi*t\
-3*pi *sin|----|
          \ 2  /
----------------
       8

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

Simplify

The third derivative [src]

     4    /pi*t\
-3*pi *cos|----|
          \ 2  /
----------------
       16

$$- \frac{3 \pi^{4} \cos{\left(\frac{\pi t}{2} \right)}}{16}$$

Simplify