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Integral of d{x}:
y(x)
y(x)
Identical expressions
y(x)=x^ two *tg(x)
y(x) equally x squared multiply by tg(x)
y(x) equally x to the power of two multiply by tg(x)
y(x)=x2*tg(x)
yx=x2*tgx
y(x)=x²*tg(x)
y(x)=x to the power of 2*tg(x)
y(x)=x^2tg(x)
y(x)=x2tg(x)
yx=x2tgx
yx=x^2tgx

The derivative of the function/ y(x)=x^2*tg(x)

Derivative of y(x)=x^2*tg(x)

The solution

You have entered [src]

 2       
x *tan(x)

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

x^2*tan(x)

Detail solution

Apply the product rule:

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

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

  /         2       2 /       2   \ /         2   \       /       2   \       \
2*\3 + 3*tan (x) + x *\1 + tan (x)/*\1 + 3*tan (x)/ + 6*x*\1 + tan (x)/*tan(x)/

$$2 \left(x^{2} \left(\tan^{2}{\left(x \right)} + 1\right) \left(3 \tan^{2}{\left(x \right)} + 1\right) + 6 x \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} + 3 \tan^{2}{\left(x \right)} + 3\right)$$

Simplify

The graph

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Identical expressions

Derivative of y(x)=x^2*tg(x)

The graph:

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The solution