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Identical expressions
(x+x^ three)*tgx
(x plus x cubed ) multiply by tgx
(x plus x to the power of three) multiply by tgx
(x+x3)*tgx
x+x3*tgx
(x+x³)*tgx
(x+x to the power of 3)*tgx
(x+x^3)tgx
(x+x3)tgx
x+x3tgx
x+x^3tgx
Similar expressions
(x-x^3)*tgx

The derivative of the function/ (x+x^3)*tgx

Derivative of (x+x^3)*tgx

The solution

You have entered [src]

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

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

d //     3\       \
--\\x + x /*tan(x)/
dx

$$\frac{d}{d x} \left(x^{3} + x\right) \tan{\left(x \right)}$$

Transform

Detail solution

Apply the product rule:

$\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$

$f{\left(x \right)} = x^{3} + x$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
1. Differentiate $x^{3} + x$ term by term:
  1. Apply the power rule: $x$ goes to $1$
  2. Apply the power rule: $x^{3}$ goes to $3 x^{2}$
  The result is: $3 x^{2} + 1$
$g{\left(x \right)} = \tan{\left(x \right)}$ ; 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: $\left(3 x^{2} + 1\right) \tan{\left(x \right)} + \frac{\left(x^{3} + x\right) \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right)}{\cos^{2}{\left(x \right)}}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

  /               /       2   \     /       2   \ /       2\            /     2\ /       2   \ /         2   \\
2*\3*tan(x) + 9*x*\1 + tan (x)/ + 3*\1 + tan (x)/*\1 + 3*x /*tan(x) + x*\1 + x /*\1 + tan (x)/*\1 + 3*tan (x)//

$$2 \left(x \left(x^{2} + 1\right) \left(\tan^{2}{\left(x \right)} + 1\right) \left(3 \tan^{2}{\left(x \right)} + 1\right) + 9 x \left(\tan^{2}{\left(x \right)} + 1\right) + 3 \cdot \left(3 x^{2} + 1\right) \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} + 3 \tan{\left(x \right)}\right)$$

Simplify

The graph

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Derivative of (x+x^3)*tgx

The graph:

Piecewise:

The solution