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Derivative of:
Derivative of (x-2)^4
Derivative of (x-2)^2*(x-4)+5
Derivative of sin^2(x/2)
Derivative of e^x*x^3
Identical expressions
(x^ three *tgx^ two)
(x cubed multiply by tgx squared )
(x to the power of three multiply by tgx to the power of two)
(x3*tgx2)
x3*tgx2
(x³*tgx²)
(x to the power of 3*tgx to the power of 2)
(x^3tgx^2)
(x3tgx2)
x3tgx2
x^3tgx^2

The derivative of the function/ (x^3*tgx^2)

Derivative of (x^3*tgx^2)

The solution

You have entered [src]

 3    2   
x *tan (x)

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

d / 3    2   \
--\x *tan (x)/
dx

$$\frac{d}{d x} x^{3} \tan^{2}{\left(x \right)}$$

Transform

Detail solution

Apply the product rule:

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

$f{\left(x \right)} = x^{3}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
1. Apply the power rule: $x^{3}$ goes to $3 x^{2}$
$g{\left(x \right)} = \tan^{2}{\left(x \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
1. Let $u = \tan{\left(x \right)}$ .
2. Apply the power rule: $u^{2}$ goes to $2 u$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \tan{\left(x \right)}$ :
  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 of the chain rule is:
  
  $\frac{2 \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \tan{\left(x \right)}}{\cos^{2}{\left(x \right)}}$
The result is: $\frac{2 x^{3} \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \tan{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 3 x^{2} \tan^{2}{\left(x \right)}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

  /     2         2 /       2   \ /         2   \        /       2   \             3 /       2   \ /         2   \       \
2*\3*tan (x) + 9*x *\1 + tan (x)/*\1 + 3*tan (x)/ + 18*x*\1 + tan (x)/*tan(x) + 4*x *\1 + tan (x)/*\2 + 3*tan (x)/*tan(x)/

$$2 \cdot \left(4 x^{3} \left(\tan^{2}{\left(x \right)} + 1\right) \left(3 \tan^{2}{\left(x \right)} + 2\right) \tan{\left(x \right)} + 9 x^{2} \left(\tan^{2}{\left(x \right)} + 1\right) \left(3 \tan^{2}{\left(x \right)} + 1\right) + 18 x \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} + 3 \tan^{2}{\left(x \right)}\right)$$

Simplify

3-я производная [src]

  /     2         2 /       2   \ /         2   \        /       2   \             3 /       2   \ /         2   \       \
2*\3*tan (x) + 9*x *\1 + tan (x)/*\1 + 3*tan (x)/ + 18*x*\1 + tan (x)/*tan(x) + 4*x *\1 + tan (x)/*\2 + 3*tan (x)/*tan(x)/

Simplify

The graph

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Identical expressions

Derivative of (x^3*tgx^2)

The graph:

Piecewise:

The solution