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Derivative of:
Derivative of (x^3-1)(x^2+x+1)
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Derivative of (x+5)/(x-1)
Derivative of x+9
Identical expressions
y=x³*(√x²)+sinx/(x²- one)
y equally x³ multiply by (√x²) plus sinus of x divide by (x² minus 1)
y equally x³ multiply by (√x²) plus sinus of x divide by (x² minus one)
y=x³(√x²)+sinx/(x²-1)
y=x³√x²+sinx/x²-1
y=x³*(√x²)+sinx divide by (x²-1)
Similar expressions
y=x³*(√x²)-sinx/(x²-1)
y=x³√x²+sinx/x²-1
y=x³(√x²)+sinx/(x²-1)
y=x³*(√x²)+sinx/(x²+1)
y=x³√x²+sinx/(x²-1)

The derivative of the function/ y=x³*(√x²)+sinx/(x²-1)

Derivative of y=x³*(√x²)+sinx/(x²-1)

The solution

You have entered [src]

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

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

x^3*(sqrt(x))^2 + sin(x)/(x^2 - 1)

Detail solution

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

                                               2       
    2    sin(x)    2*sin(x)    4*x*cos(x)   8*x *sin(x)
12*x  - ------- - ---------- - ---------- + -----------
              2            2            2             3
        -1 + x    /      2\    /      2\     /      2\ 
                  \-1 + x /    \-1 + x /     \-1 + x /

$$12 x^{2} + \frac{8 x^{2} \sin{\left(x \right)}}{\left(x^{2} - 1\right)^{3}} - \frac{4 x \cos{\left(x \right)}}{\left(x^{2} - 1\right)^{2}} - \frac{\sin{\left(x \right)}}{x^{2} - 1} - \frac{2 \sin{\left(x \right)}}{\left(x^{2} - 1\right)^{2}}$$

Simplify

The third derivative [src]

                                  3                                         2       
        cos(x)    6*cos(x)    48*x *sin(x)   6*x*sin(x)   24*x*sin(x)   24*x *cos(x)
24*x - ------- - ---------- - ------------ + ---------- + ----------- + ------------
             2            2             4             2             3             3 
       -1 + x    /      2\     /      2\     /      2\     /      2\     /      2\  
                 \-1 + x /     \-1 + x /     \-1 + x /     \-1 + x /     \-1 + x /

$$- \frac{48 x^{3} \sin{\left(x \right)}}{\left(x^{2} - 1\right)^{4}} + \frac{24 x^{2} \cos{\left(x \right)}}{\left(x^{2} - 1\right)^{3}} + 24 x + \frac{6 x \sin{\left(x \right)}}{\left(x^{2} - 1\right)^{2}} + \frac{24 x \sin{\left(x \right)}}{\left(x^{2} - 1\right)^{3}} - \frac{\cos{\left(x \right)}}{x^{2} - 1} - \frac{6 \cos{\left(x \right)}}{\left(x^{2} - 1\right)^{2}}$$

Simplify

The graph

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Derivative of y=x³*(√x²)+sinx/(x²-1)

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The solution