Mister Exam

Lang:

How to use it?
Derivative of:
Derivative of (x^5+1)
Derivative of x^sqrt(x)
Derivative of log(x+2)
Derivative of ex^2
Identical expressions
x*sin(pi*x/ two ^(- two))
x multiply by sinus of ( Pi multiply by x divide by 2 to the power of ( minus 2))
x multiply by sinus of ( Pi multiply by x divide by two to the power of ( minus two))
x*sin(pi*x/2(-2))
x*sinpi*x/2-2
xsin(pix/2^(-2))
xsin(pix/2(-2))
xsinpix/2-2
xsinpix/2^-2
x*sin(pi*x divide by 2^(-2))
Similar expressions
x*sin(pi*x/2^(2))

Derivative of xsin(pix/2^(-2))

You have entered [src]

     /pi*x\
x*sin|----|
     \0.25/

$$x \sin{\left(\frac{\pi x}{0.25} \right)}$$

x*sin((pi*x)/0.25)

Detail solution

Apply the product rule:

$\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$

$f{\left(x \right)} = x$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
1. Apply the power rule: $x$ goes to $1$
$g{\left(x \right)} = \sin{\left(\frac{\pi x}{0.25} \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
1. Let $u = \frac{\pi x}{0.25}$ .
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 $\frac{d}{d x} \frac{\pi x}{0.25}$ :
  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: $4.0 \pi$
  The result of the chain rule is:
  
  $4.0 \pi \cos{\left(\frac{\pi x}{0.25} \right)}$
The result is: $4.0 \pi x \cos{\left(\frac{\pi x}{0.25} \right)} + \sin{\left(\frac{\pi x}{0.25} \right)}$
Now simplify:

$4.0 \pi x \cos{\left(4 \pi x \right)} + \sin{\left(4 \pi x \right)}$

The answer is:

$4.0 \pi x \cos{\left(4 \pi x \right)} + \sin{\left(4 \pi x \right)}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

            /pi*x\      /pi*x\
4.0*pi*x*cos|----| + sin|----|
            \0.25/      \0.25/

$$4.0 \pi x \cos{\left(\frac{\pi x}{0.25} \right)} + \sin{\left(\frac{\pi x}{0.25} \right)}$$

Simplify

The second derivative [src]

pi*(8.0*cos(4*pi*x) - 16.0*pi*x*sin(4*pi*x))

$$\pi \left(- 16.0 \pi x \sin{\left(4 \pi x \right)} + 8.0 \cos{\left(4 \pi x \right)}\right)$$

Simplify

The third derivative [src]

   2                                           
-pi *(48.0*sin(4*pi*x) + 64.0*pi*x*cos(4*pi*x))

$$- \pi^{2} \left(64.0 \pi x \cos{\left(4 \pi x \right)} + 48.0 \sin{\left(4 \pi x \right)}\right)$$

Simplify

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