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Similar expressions
sin(x)-(1/(1-x^2))
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The derivative of the function/ sin(x)-(1/(1+x^2))

Derivative of sin(x)-(1/(1+x^2))

The solution

You have entered [src]

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

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

sin(x) - 1/(1 + x^2)

Detail solution

Differentiate $\sin{\left(x \right)} - \frac{1}{x^{2} + 1}$ term by term:
1. The derivative of sine is cosine:
  
  $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
2. The derivative of a constant times a function is the constant times the derivative of the function.
  1. Let $u = x^{2} + 1$ .
  2. Apply the power rule: $\frac{1}{u}$ goes to $- \frac{1}{u^{2}}$
  3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(x^{2} + 1\right)$ :
    1. Differentiate $x^{2} + 1$ term by term:
      1. The derivative of the constant $1$ is zero.
      2. Apply the power rule: $x^{2}$ goes to $2 x$
      The result is: $2 x$
    The result of the chain rule is:
    
    $- \frac{2 x}{\left(x^{2} + 1\right)^{2}}$
  So, the result is: $\frac{2 x}{\left(x^{2} + 1\right)^{2}}$
The result is: $\frac{2 x}{\left(x^{2} + 1\right)^{2}} + \cos{\left(x \right)}$

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

   2*x            
--------- + cos(x)
        2         
/     2\          
\1 + x /

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

Simplify

The second derivative [src]

                            2  
              2          8*x   
-sin(x) + --------- - ---------
                  2           3
          /     2\    /     2\ 
          \1 + x /    \1 + x /

$$- \frac{8 x^{2}}{\left(x^{2} + 1\right)^{3}} - \sin{\left(x \right)} + \frac{2}{\left(x^{2} + 1\right)^{2}}$$

Simplify

The third derivative [src]

                            3  
             24*x       48*x   
-cos(x) - --------- + ---------
                  3           4
          /     2\    /     2\ 
          \1 + x /    \1 + x /

$$\frac{48 x^{3}}{\left(x^{2} + 1\right)^{4}} - \frac{24 x}{\left(x^{2} + 1\right)^{3}} - \cos{\left(x \right)}$$

Simplify

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Derivative of sin(x)-(1/(1+x^2))

The graph:

Piecewise:

The solution