Mister Exam

Lang:

Other calculators

How to use it?
Derivative of:
Derivative of e^-1
Derivative of 2*x^5
Derivative of f(x)=1
Derivative of (x+8)^2
Identical expressions
two *sin(pi*x/ four)+ three
2 multiply by sinus of ( Pi multiply by x divide by 4) plus 3
two multiply by sinus of ( Pi multiply by x divide by four) plus three
2sin(pix/4)+3
2sinpix/4+3
2*sin(pi*x divide by 4)+3
Similar expressions
2*sin(pi*x/4)-3

The derivative of the function/ 2*sin(pi*x/4)+3

Derivative of 2sin(pix/4)+3

The solution

You have entered [src]

     /pi*x\    
2*sin|----| + 3
     \ 4  /

$$2 \sin{\left(\frac{\pi x}{4} \right)} + 3$$

2*sin((pi*x)/4) + 3

Detail solution

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

$\frac{\pi \cos{\left(\frac{\pi x}{4} \right)}}{2}$

The answer is:

$\frac{\pi \cos{\left(\frac{\pi x}{4} \right)}}{2}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

      /pi*x\
pi*cos|----|
      \ 4  /
------------
     2

$$\frac{\pi \cos{\left(\frac{\pi x}{4} \right)}}{2}$$

Simplify

The second derivative [src]

   2    /pi*x\ 
-pi *sin|----| 
        \ 4  / 
---------------
       8

$$- \frac{\pi^{2} \sin{\left(\frac{\pi x}{4} \right)}}{8}$$

Simplify

The third derivative [src]

   3    /pi*x\ 
-pi *cos|----| 
        \ 4  / 
---------------
       32

$$- \frac{\pi^{3} \cos{\left(\frac{\pi x}{4} \right)}}{32}$$

Simplify