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The derivative of the function/ sqrt(2-3x^2+5x)

Derivative of sqrt(2-3x^2+5x)

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

You have entered [src] 
   ________________
  /        2       
\/  2 - 3*x  + 5*x
5x+(2−3x2)\sqrt{5 x + \left(2 - 3 x^{2}\right)} 
sqrt(2 - 3*x^2 + 5*x)
Detail solution

  1. Let u=5x+(2−3x2)u = 5 x + \left(2 - 3 x^{2}\right).

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

  3. Then, apply the chain rule. Multiply by ddx(5x+(2−3x2))\frac{d}{d x} \left(5 x + \left(2 - 3 x^{2}\right)\right):

    1. Differentiate 5x+(2−3x2)5 x + \left(2 - 3 x^{2}\right) term by term:

      1. Differentiate 2−3x22 - 3 x^{2} term by term:

        1. The derivative of the constant 22 is zero.

        2. The derivative of a constant times a function is the constant times the derivative of the function.

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

          So, the result is: −6x- 6 x

        The result is: −6x- 6 x

      2. The derivative of a constant times a function is the constant times the derivative of the function.

        1. Apply the power rule: xx goes to 11

        So, the result is: 55

      The result is: 5−6x5 - 6 x

    The result of the chain rule is:

    5−6x25x+(2−3x2)\frac{5 - 6 x}{2 \sqrt{5 x + \left(2 - 3 x^{2}\right)}}

  4. Now simplify:

    5−6x2−3x2+5x+2\frac{5 - 6 x}{2 \sqrt{- 3 x^{2} + 5 x + 2}}

The answer is:

5−6x2−3x2+5x+2\frac{5 - 6 x}{2 \sqrt{- 3 x^{2} + 5 x + 2}}

The graph
02468-8-6-4-2-1010-1010
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
     5/2 - 3*x     
-------------------
   ________________
  /        2       
\/  2 - 3*x  + 5*x
52−3x5x+(2−3x2)\frac{\frac{5}{2} - 3 x}{\sqrt{5 x + \left(2 - 3 x^{2}\right)}}
Simplify
The second derivative [src] 
 /                 2    \ 
 |       (-5 + 6*x)     | 
-|3 + ------------------| 
 |      /       2      \| 
 \    4*\2 - 3*x  + 5*x// 
--------------------------
      ________________    
     /        2           
   \/  2 - 3*x  + 5*x
−(6x−5)24(−3x2+5x+2)+3−3x2+5x+2- \frac{\frac{\left(6 x - 5\right)^{2}}{4 \left(- 3 x^{2} + 5 x + 2\right)} + 3}{\sqrt{- 3 x^{2} + 5 x + 2}}
Simplify
The third derivative [src] 
              /                2  \
              |      (-5 + 6*x)   |
-3*(-5 + 6*x)*|12 + --------------|
              |            2      |
              \     2 - 3*x  + 5*x/
-----------------------------------
                         3/2       
         /       2      \          
       8*\2 - 3*x  + 5*x/
−3(6x−5)((6x−5)2−3x2+5x+2+12)8(−3x2+5x+2)32- \frac{3 \left(6 x - 5\right) \left(\frac{\left(6 x - 5\right)^{2}}{- 3 x^{2} + 5 x + 2} + 12\right)}{8 \left(- 3 x^{2} + 5 x + 2\right)^{\frac{3}{2}}}
Simplify
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