Derivative of y=sinx/(2*x)+1

You have entered [src]

sin(x)    
------ + 1
 2*x

$$1 + \frac{\sin{\left(x \right)}}{2 x}$$

sin(x)/((2*x)) + 1

Detail solution

Differentiate $1 + \frac{\sin{\left(x \right)}}{2 x}$ term by term:
1. Apply the quotient rule, which is:
  
  $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
  
  $f{\left(x \right)} = \sin{\left(x \right)}$ and $g{\left(x \right)} = 2 x$ .
  
  To find $\frac{d}{d x} f{\left(x \right)}$ :
  1. The derivative of sine is cosine:
    
    $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
  To find $\frac{d}{d x} g{\left(x \right)}$ :
  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: $2$
  Now plug in to the quotient rule:
  
  $\frac{2 x \cos{\left(x \right)} - 2 \sin{\left(x \right)}}{4 x^{2}}$
2. The derivative of the constant $1$ is zero.
The result is: $\frac{2 x \cos{\left(x \right)} - 2 \sin{\left(x \right)}}{4 x^{2}}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

$$\frac{1}{2 x} \cos{\left(x \right)} - \frac{\sin{\left(x \right)}}{2 x^{2}}$$

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

The graph

Other calculators

How to use it?

Identical expressions

Similar expressions

Derivative of y=sinx/(2*x)+1

The graph:

Piecewise:

The solution