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Derivative of x^12
Derivative of tanx
Derivative of tg2x
Derivative of 3/5*x
Identical expressions
y=sqrt(x+ two sqrt(x))+(three)/(5x^(two)+2x- one)^2
y equally square root of (x plus 2 square root of (x)) plus (3) divide by (5x to the power of (2) plus 2x minus 1) squared
y equally square root of (x plus two square root of (x)) plus (three) divide by (5x to the power of (two) plus 2x minus one) squared
y=√(x+2√(x))+(3)/(5x^(2)+2x-1)^2
y=sqrt(x+2sqrt(x))+(3)/(5x(2)+2x-1)2
y=sqrtx+2sqrtx+3/5x2+2x-12
y=sqrt(x+2sqrt(x))+(3)/(5x^(2)+2x-1)²
y=sqrt(x+2sqrt(x))+(3)/(5x to the power of (2)+2x-1) to the power of 2
y=sqrtx+2sqrtx+3/5x^2+2x-1^2
y=sqrt(x+2sqrt(x))+(3) divide by (5x^(2)+2x-1)^2
Similar expressions
y=sqrt(x+2sqrt(x))-(3)/(5x^(2)+2x-1)^2
y=sqrt(x-2sqrt(x))+(3)/(5x^(2)+2x-1)^2
y=sqrt(x+2sqrt(x))+(3)/(5x^(2)+2x+1)^2
y=sqrt(x+2sqrt(x))+(3)/(5x^(2)-2x-1)^2

The derivative of the function/ y=sqrt(x+2sqrt(x))+(3)/(5x^(2)+2x-1)^2

Derivative of y=sqrt(x+2sqrt(x))+(3)/(5x^(2)+2x-1)^2

The solution

You have entered [src]

   _____________                    
  /         ___            3        
\/  x + 2*\/ x   + -----------------
                                   2
                   /   2          \ 
                   \5*x  + 2*x - 1/

$$\sqrt{2 \sqrt{x} + x} + \frac{3}{\left(\left(5 x^{2} + 2 x\right) - 1\right)^{2}}$$

sqrt(x + 2*sqrt(x)) + 3/(5*x^2 + 2*x - 1)^2

Detail solution

Differentiate $\sqrt{2 \sqrt{x} + x} + \frac{3}{\left(\left(5 x^{2} + 2 x\right) - 1\right)^{2}}$ term by term:
1. Let $u = 2 \sqrt{x} + x$ .
2. Apply the power rule: $\sqrt{u}$ goes to $\frac{1}{2 \sqrt{u}}$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(2 \sqrt{x} + x\right)$ :
  1. Differentiate $2 \sqrt{x} + x$ term by term:
    1. Apply the power rule: $x$ goes to $1$
    2. The derivative of a constant times a function is the constant times the derivative of the function.
      1. Apply the power rule: $\sqrt{x}$ goes to $\frac{1}{2 \sqrt{x}}$
      So, the result is: $\frac{1}{\sqrt{x}}$
    The result is: $1 + \frac{1}{\sqrt{x}}$
  The result of the chain rule is:
  
  $\frac{1 + \frac{1}{\sqrt{x}}}{2 \sqrt{2 \sqrt{x} + x}}$
4. The derivative of a constant times a function is the constant times the derivative of the function.
  1. Let $u = \left(\left(5 x^{2} + 2 x\right) - 1\right)^{2}$ .
  2. Apply the power rule: $\frac{1}{u}$ goes to $- \frac{1}{u^{2}}$
  3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(\left(5 x^{2} + 2 x\right) - 1\right)^{2}$ :
    1. Let $u = \left(5 x^{2} + 2 x\right) - 1$ .
    2. Apply the power rule: $u^{2}$ goes to $2 u$
    3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(\left(5 x^{2} + 2 x\right) - 1\right)$ :
      1. Differentiate $\left(5 x^{2} + 2 x\right) - 1$ term by term:
        
        Differentiate $5 x^{2} + 2 x$ term by term:
        
        The derivative of a constant times a function is the constant times the derivative of the function.
        
        Apply the power rule: $x^{2}$ goes to $2 x$
        
        So, the result is: $10 x$
        
        The derivative of a constant times a function is the constant times the derivative of the function.
        
        Apply the power rule: $x$ goes to $1$
        
        So, the result is: $2$
        
        The result is: $10 x + 2$
        
        The derivative of the constant $-1$ is zero.
        
        The result is: $10 x + 2$
      The result of the chain rule is:
      
      $\left(10 x + 2\right) \left(2 \left(5 x^{2} + 2 x\right) - 2\right)$
    The result of the chain rule is:
    
    $- \frac{\left(10 x + 2\right) \left(2 \left(5 x^{2} + 2 x\right) - 2\right)}{\left(\left(5 x^{2} + 2 x\right) - 1\right)^{4}}$
  So, the result is: $- \frac{3 \left(10 x + 2\right) \left(2 \left(5 x^{2} + 2 x\right) - 2\right)}{\left(\left(5 x^{2} + 2 x\right) - 1\right)^{4}}$
The result is: $\frac{1 + \frac{1}{\sqrt{x}}}{2 \sqrt{2 \sqrt{x} + x}} - \frac{3 \left(10 x + 2\right) \left(2 \left(5 x^{2} + 2 x\right) - 2\right)}{\left(\left(5 x^{2} + 2 x\right) - 1\right)^{4}}$
Now simplify:

$- \frac{60 x}{\left(5 x^{2} + 2 x - 1\right)^{3}} - \frac{12}{\left(5 x^{2} + 2 x - 1\right)^{3}} + \frac{1}{2 \sqrt{2 \sqrt{x} + x}} + \frac{1}{2 \sqrt{x} \sqrt{2 \sqrt{x} + x}}$

The answer is:

$- \frac{60 x}{\left(5 x^{2} + 2 x - 1\right)^{3}} - \frac{12}{\left(5 x^{2} + 2 x - 1\right)^{3}} + \frac{1}{2 \sqrt{2 \sqrt{x} + x}} + \frac{1}{2 \sqrt{x} \sqrt{2 \sqrt{x} + x}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

  1      1                          
  - + -------                       
  2       ___                       
      2*\/ x          3*(4 + 20*x)  
---------------- - -----------------
   _____________                   3
  /         ___    /   2          \ 
\/  x + 2*\/ x     \5*x  + 2*x - 1/

$$\frac{\frac{1}{2} + \frac{1}{2 \sqrt{x}}}{\sqrt{2 \sqrt{x} + x}} - \frac{3 \left(20 x + 4\right)}{\left(\left(5 x^{2} + 2 x\right) - 1\right)^{3}}$$

Simplify

The second derivative [src]

                                                                                    2   
                                                                         /      1  \    
                                                                         |1 + -----|    
                                     2                                   |      ___|    
          60             72*(1 + 5*x)                  1                 \    \/ x /    
- ------------------ + ------------------ - ----------------------- - ------------------
                   3                    4             _____________                  3/2
  /              2\    /              2\       3/2   /         ___      /        ___\   
  \-1 + 2*x + 5*x /    \-1 + 2*x + 5*x /    4*x   *\/  x + 2*\/ x     4*\x + 2*\/ x /

$$- \frac{\left(1 + \frac{1}{\sqrt{x}}\right)^{2}}{4 \left(2 \sqrt{x} + x\right)^{\frac{3}{2}}} + \frac{72 \left(5 x + 1\right)^{2}}{\left(5 x^{2} + 2 x - 1\right)^{4}} - \frac{60}{\left(5 x^{2} + 2 x - 1\right)^{3}} - \frac{1}{4 x^{\frac{3}{2}} \sqrt{2 \sqrt{x} + x}}$$

Simplify

The third derivative [src]

  /                                                                                    3                             \
  |                                                                         /      1  \                    1         |
  |                                                                         |1 + -----|              1 + -----       |
  |                 3                                                       |      ___|                    ___       |
  |    192*(1 + 5*x)        360*(1 + 5*x)                 1                 \    \/ x /                  \/ x        |
3*|- ------------------ + ------------------ + ----------------------- + ------------------ + -----------------------|
  |                   5                    4             _____________                  5/2                       3/2|
  |  /              2\    /              2\       5/2   /         ___      /        ___\         3/2 /        ___\   |
  \  \-1 + 2*x + 5*x /    \-1 + 2*x + 5*x /    8*x   *\/  x + 2*\/ x     8*\x + 2*\/ x /      8*x   *\x + 2*\/ x /   /

$$3 \left(\frac{\left(1 + \frac{1}{\sqrt{x}}\right)^{3}}{8 \left(2 \sqrt{x} + x\right)^{\frac{5}{2}}} - \frac{192 \left(5 x + 1\right)^{3}}{\left(5 x^{2} + 2 x - 1\right)^{5}} + \frac{360 \left(5 x + 1\right)}{\left(5 x^{2} + 2 x - 1\right)^{4}} + \frac{1 + \frac{1}{\sqrt{x}}}{8 x^{\frac{3}{2}} \left(2 \sqrt{x} + x\right)^{\frac{3}{2}}} + \frac{1}{8 x^{\frac{5}{2}} \sqrt{2 \sqrt{x} + x}}\right)$$

Simplify

The graph

Derivative of y=sqrt(x+2sqrt(x))+(3)/(5x^(2)+2x-1)^2

Other calculators

How to use it?

Identical expressions

Similar expressions

Derivative of y=sqrt(x+2sqrt(x))+(3)/(5x^(2)+2x-1)^2

The graph:

Piecewise:

The solution