Mister Exam

Lang:

Other calculators

How to use it?
Derivative of:
Derivative of x*(log(x)/log(10))
Derivative of x/5
Derivative of x^2*e^(-x)
Derivative of -(x^2+1)/x
Identical expressions
y=sinx/ two *x+ sixteen
y equally sinus of x divide by 2 multiply by x plus 16
y equally sinus of x divide by two multiply by x plus sixteen
y=sinx/2x+16
y=sinx divide by 2*x+16
Similar expressions
y=sinx/2*x-16

The derivative of the function/ y=sinx/2*x+16

Derivative of y=sinx/2*x+16

The solution

You have entered [src]

sin(x)       
------*x + 16
  2

$$x \frac{\sin{\left(x \right)}}{2} + 16$$

(sin(x)/2)*x + 16

Detail solution

Differentiate $x \frac{\sin{\left(x \right)}}{2} + 16$ term by term:
1. Apply the quotient rule, which is:
  
  $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
  
  $f{\left(x \right)} = x \sin{\left(x \right)}$ and $g{\left(x \right)} = 2$ .
  
  To find $\frac{d}{d x} f{\left(x \right)}$ :
  1. 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 $1$
    $g{\left(x \right)} = \sin{\left(x \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
    1. The derivative of sine is cosine:
      
      $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
    The result is: $x \cos{\left(x \right)} + \sin{\left(x \right)}$
  To find $\frac{d}{d x} g{\left(x \right)}$ :
  1. The derivative of the constant $2$ is zero.
  Now plug in to the quotient rule:
  
  $\frac{x \cos{\left(x \right)}}{2} + \frac{\sin{\left(x \right)}}{2}$
2. The derivative of the constant $16$ is zero.
The result is: $\frac{x \cos{\left(x \right)}}{2} + \frac{\sin{\left(x \right)}}{2}$

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

-(3*sin(x) + x*cos(x)) 
-----------------------
           2

$$- \frac{x \cos{\left(x \right)} + 3 \sin{\left(x \right)}}{2}$$

Simplify

The graph

Derivative of y=sinx/2*x+16