Derivative of xtgx/1+x²

The solution

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

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

d /x*tan(x)    2\
--|-------- + x |
dx\   1         /

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

Transform

Detail solution

Differentiate $x^{2} + \frac{x \tan{\left(x \right)}}{1}$ term by term:
1. The derivative of a constant times a function is the constant times the derivative of the function.
  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 $1$
    $g{\left(x \right)} = \tan{\left(x \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
    1. Rewrite the function to be differentiated:
      
      $\tan{\left(x \right)} = \frac{\sin{\left(x \right)}}{\cos{\left(x \right)}}$
    2. 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 $g{\left(x \right)} = \cos{\left(x \right)}$ .
      
      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 cosine is negative sine:
        
        $\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}$
      Now plug in to the quotient rule:
      
      $\frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}}$
    The result is: $\frac{x \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right)}{\cos^{2}{\left(x \right)}} + \tan{\left(x \right)}$
  So, the result is: $\frac{x \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right)}{\cos^{2}{\left(x \right)}} + \tan{\left(x \right)}$
2. Apply the power rule: $x^{2}$ goes to $2 x$
The result is: $\frac{x \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right)}{\cos^{2}{\left(x \right)}} + 2 x + \tan{\left(x \right)}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

        /       2   \         
2*x + x*\1 + tan (x)/ + tan(x)

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

Simplify

The second derivative [src]

  /       2        /       2   \       \
2*\2 + tan (x) + x*\1 + tan (x)/*tan(x)/

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

Simplify

The third derivative [src]

  /       2   \ /             /       2   \          2   \
2*\1 + tan (x)/*\3*tan(x) + x*\1 + tan (x)/ + 2*x*tan (x)/

$$2 \left(\tan^{2}{\left(x \right)} + 1\right) \left(x \left(\tan^{2}{\left(x \right)} + 1\right) + 2 x \tan^{2}{\left(x \right)} + 3 \tan{\left(x \right)}\right)$$

Simplify

The graph

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Derivative of xtgx/1+x²

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The solution