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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)

Function f() - derivative -N order at the point
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The solution

You have entered [src] 
        2         
 3   ___    sin(x)
x *\/ x   + ------
             2    
            x  - 1
x3(x)2+sin⁡(x)x2−1x^{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

  1. Differentiate x3(x)2+sin⁡(x)x2−1x^{3} \left(\sqrt{x}\right)^{2} + \frac{\sin{\left(x \right)}}{x^{2} - 1} term by term:

    1. Apply the product rule:

      ddxf(x)g(x)=f(x)ddxg(x)+g(x)ddxf(x)\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(x)=x3f{\left(x \right)} = x^{3}; to find ddxf(x)\frac{d}{d x} f{\left(x \right)}:

      1. Apply the power rule: x3x^{3} goes to 3x23 x^{2}

      g(x)=(x)2g{\left(x \right)} = \left(\sqrt{x}\right)^{2}; to find ddxg(x)\frac{d}{d x} g{\left(x \right)}:

      1. Let u=xu = \sqrt{x}.

      2. Apply the power rule: u2u^{2} goes to 2u2 u

      3. Then, apply the chain rule. Multiply by ddxx\frac{d}{d x} \sqrt{x}:

        1. Apply the power rule: x\sqrt{x} goes to 12x\frac{1}{2 \sqrt{x}}

        The result of the chain rule is:

        11

      The result is: x3+3xx2x^{3} + 3 x x^{2}

    2. Apply the quotient rule, which is:

      ddxf(x)g(x)=−f(x)ddxg(x)+g(x)ddxf(x)g2(x)\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(x)=sin⁡(x)f{\left(x \right)} = \sin{\left(x \right)} and g(x)=x2−1g{\left(x \right)} = x^{2} - 1.

      To find ddxf(x)\frac{d}{d x} f{\left(x \right)}:

      1. The derivative of sine is cosine:

        ddxsin⁡(x)=cos⁡(x)\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}

      To find ddxg(x)\frac{d}{d x} g{\left(x \right)}:

      1. Differentiate x2−1x^{2} - 1 term by term:

        1. The derivative of the constant −1-1 is zero.

        2. Apply the power rule: x2x^{2} goes to 2x2 x

        The result is: 2x2 x

      Now plug in to the quotient rule:

      −2xsin⁡(x)+(x2−1)cos⁡(x)(x2−1)2\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: x3+3xx2+−2xsin⁡(x)+(x2−1)cos⁡(x)(x2−1)2x^{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}}

  2. Now simplify:

    4x3(x2−1)2−2xsin⁡(x)+(x2−1)cos⁡(x)(x2−1)2\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:

4x3(x2−1)2−2xsin⁡(x)+(x2−1)cos⁡(x)(x2−1)2\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
02468-8-6-4-2-1010-2000020000
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/
x3+3xx2−2xsin⁡(x)(x2−1)2+cos⁡(x)x2−1x^{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 /
12x2+8x2sin⁡(x)(x2−1)3−4xcos⁡(x)(x2−1)2−sin⁡(x)x2−1−2sin⁡(x)(x2−1)212 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 /
−48x3sin⁡(x)(x2−1)4+24x2cos⁡(x)(x2−1)3+24x+6xsin⁡(x)(x2−1)2+24xsin⁡(x)(x2−1)3−cos⁡(x)x2−1−6cos⁡(x)(x2−1)2- \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
Derivative of y=x³*(√x²)+sinx/(x²-1)
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