Derivative of y=x³√x²+sinx/x²-1

The solution

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

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

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

Detail solution

Differentiate $\left(x^{3} \left(\sqrt{x}\right)^{2} + \frac{\sin{\left(x \right)}}{x^{2}}\right) - 1$ term by term:
1. Differentiate $x^{3} \left(\sqrt{x}\right)^{2} + \frac{\sin{\left(x \right)}}{x^{2}}$ 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 $g{\left(x \right)} = x^{2}$ .
    
    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. Apply the power rule: $x^{2}$ goes to $2 x$
    Now plug in to the quotient rule:
    
    $\frac{x^{2} \cos{\left(x \right)} - 2 x \sin{\left(x \right)}}{x^{4}}$
  The result is: $x^{3} + 3 x x^{2} + \frac{x^{2} \cos{\left(x \right)} - 2 x \sin{\left(x \right)}}{x^{4}}$
2. The derivative of the constant $-1$ is zero.
The result is: $x^{3} + 3 x x^{2} + \frac{x^{2} \cos{\left(x \right)} - 2 x \sin{\left(x \right)}}{x^{4}}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

 3   cos(x)   2*sin(x)        2
x  + ------ - -------- + 3*x*x 
        2         3            
       x         x

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

Simplify

The second derivative [src]

    2   sin(x)   4*cos(x)   6*sin(x)
12*x  - ------ - -------- + --------
           2         3          4   
          x         x          x

$$12 x^{2} - \frac{\sin{\left(x \right)}}{x^{2}} - \frac{4 \cos{\left(x \right)}}{x^{3}} + \frac{6 \sin{\left(x \right)}}{x^{4}}$$

Simplify

The third derivative [src]

       cos(x)   24*sin(x)   6*sin(x)   18*cos(x)
24*x - ------ - --------- + -------- + ---------
          2          5          3           4   
         x          x          x           x

$$24 x - \frac{\cos{\left(x \right)}}{x^{2}} + \frac{6 \sin{\left(x \right)}}{x^{3}} + \frac{18 \cos{\left(x \right)}}{x^{4}} - \frac{24 \sin{\left(x \right)}}{x^{5}}$$

Simplify

The graph

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Derivative of y=x³√x²+sinx/x²-1

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The solution