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Derivative of:
Derivative of x^(1/5)
Derivative of x^-3
Derivative of e^(4*x)
Derivative of (x+4)^2
Identical expressions
sqrt(two -3x^ two +5x)
square root of (2 minus 3x squared plus 5x)
square root of (two minus 3x to the power of two plus 5x)
√(2-3x^2+5x)
sqrt(2-3x2+5x)
sqrt2-3x2+5x
sqrt(2-3x²+5x)
sqrt(2-3x to the power of 2+5x)
sqrt2-3x^2+5x
Similar expressions
sqrt(2+3x^2+5x)
sqrt(2-3x^2-5x)

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

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

The solution

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   ________________
  /        2       
\/  2 - 3*x  + 5*x

$$\sqrt{5 x + \left(2 - 3 x^{2}\right)}$$

sqrt(2 - 3*x^2 + 5*x)

Detail solution

Let $u = 5 x + \left(2 - 3 x^{2}\right)$ .
Apply the power rule: $\sqrt{u}$ goes to $\frac{1}{2 \sqrt{u}}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(5 x + \left(2 - 3 x^{2}\right)\right)$ :
1. Differentiate $5 x + \left(2 - 3 x^{2}\right)$ term by term:
  1. Differentiate $2 - 3 x^{2}$ term by term:
    1. The derivative of the constant $2$ 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: $x^{2}$ goes to $2 x$
      So, the result is: $- 6 x$
    The result is: $- 6 x$
  2. The derivative of a constant times a function is the constant times the derivative of the function.
    1. Apply the power rule: $x$ goes to $1$
    So, the result is: $5$
  The result is: $5 - 6 x$
The result of the chain rule is:

$\frac{5 - 6 x}{2 \sqrt{5 x + \left(2 - 3 x^{2}\right)}}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

     5/2 - 3*x     
-------------------
   ________________
  /        2       
\/  2 - 3*x  + 5*x

$$\frac{\frac{5}{2} - 3 x}{\sqrt{5 x + \left(2 - 3 x^{2}\right)}}$$

Simplify

The second derivative [src]

 /                 2    \ 
 |       (-5 + 6*x)     | 
-|3 + ------------------| 
 |      /       2      \| 
 \    4*\2 - 3*x  + 5*x// 
--------------------------
      ________________    
     /        2           
   \/  2 - 3*x  + 5*x

$$- \frac{\frac{\left(6 x - 5\right)^{2}}{4 \left(- 3 x^{2} + 5 x + 2\right)} + 3}{\sqrt{- 3 x^{2} + 5 x + 2}}$$

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/

$$- \frac{3 \left(6 x - 5\right) \left(\frac{\left(6 x - 5\right)^{2}}{- 3 x^{2} + 5 x + 2} + 12\right)}{8 \left(- 3 x^{2} + 5 x + 2\right)^{\frac{3}{2}}}$$

Simplify

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

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The solution