Mister Exam

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Derivative of:
Derivative of (x+5)/(x+1)
Derivative of x^(4/3)
Derivative of x*2^x
Derivative of x+log(x)
Identical expressions
x*sin(pi*x/ two ^(- one))
x multiply by sinus of ( Pi multiply by x divide by 2 to the power of ( minus 1))
x multiply by sinus of ( Pi multiply by x divide by two to the power of ( minus one))
x*sin(pi*x/2(-1))
x*sinpi*x/2-1
xsin(pix/2^(-1))
xsin(pix/2(-1))
xsinpix/2-1
xsinpix/2^-1
x*sin(pi*x divide by 2^(-1))
Similar expressions
x*sin(pi*x/2^(1))

Derivative of xsin(pix/2^(-1))

You have entered [src]

     /pi*x\
x*sin|----|
     \0.5 /

$$x \sin{\left(\frac{\pi x}{0.5} \right)}$$

x*sin((pi*x)/0.5)

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 $1$
$g{\left(x \right)} = \sin{\left(\frac{\pi x}{0.5} \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
1. Let $u = \frac{\pi x}{0.5}$ .
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 $\frac{d}{d x} \frac{\pi x}{0.5}$ :
  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: $2.0 \pi$
  The result of the chain rule is:
  
  $2.0 \pi \cos{\left(\frac{\pi x}{0.5} \right)}$
The result is: $2.0 \pi x \cos{\left(\frac{\pi x}{0.5} \right)} + \sin{\left(\frac{\pi x}{0.5} \right)}$
Now simplify:

$2.0 \pi x \cos{\left(2 \pi x \right)} + \sin{\left(2 \pi x \right)}$

The answer is:

$2.0 \pi x \cos{\left(2 \pi x \right)} + \sin{\left(2 \pi x \right)}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

            /pi*x\      /pi*x\
2.0*pi*x*cos|----| + sin|----|
            \0.5 /      \0.5 /

$$2.0 \pi x \cos{\left(\frac{\pi x}{0.5} \right)} + \sin{\left(\frac{\pi x}{0.5} \right)}$$

Simplify

The second derivative [src]

4.0*pi*(-pi*x*sin(2*pi*x) + cos(2*pi*x))

$$4.0 \pi \left(- \pi x \sin{\left(2 \pi x \right)} + \cos{\left(2 \pi x \right)}\right)$$

Simplify

The third derivative [src]

   2                                          
-pi *(12.0*sin(2*pi*x) + 8.0*pi*x*cos(2*pi*x))

$$- \pi^{2} \left(8.0 \pi x \cos{\left(2 \pi x \right)} + 12.0 \sin{\left(2 \pi x \right)}\right)$$

Simplify

Derivative of x*sin(pi*x/2^(-1))