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Identical expressions
y=lnsqrt(two x-5x^2+ one)
y equally ln square root of (2x minus 5x squared plus 1)
y equally ln square root of (two x minus 5x squared plus one)
y=ln√(2x-5x^2+1)
y=lnsqrt(2x-5x2+1)
y=lnsqrt2x-5x2+1
y=lnsqrt(2x-5x²+1)
y=lnsqrt(2x-5x to the power of 2+1)
y=lnsqrt2x-5x^2+1
Similar expressions
y=lnsqrt(2x-5x^2-1)
y=lnsqrt(2x+5x^2+1)

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

Derivative of y=lnsqrt(2x-5x^2+1)

The solution

You have entered [src]

   /   ________________\
   |  /          2     |
log\\/  2*x - 5*x  + 1 /

\log{\left(\sqrt{\left(- 5 x^{2} + 2 x\right) + 1} \right)}

log(sqrt(2*x - 5*x^2 + 1))

Detail solution

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

$\frac{2 - 10 x}{2 \left(\left(- 5 x^{2} + 2 x\right) + 1\right)}$
Now simplify:

$\frac{1 - 5 x}{- 5 x^{2} + 2 x + 1}$

The answer is:

\frac{1 - 5 x}{- 5 x^{2} + 2 x + 1}

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

   1 - 5*x    
--------------
         2    
2*x - 5*x  + 1

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

Simplify

The second derivative [src]

 /                2 \ 
 |    2*(-1 + 5*x)  | 
-|5 + --------------| 
 |           2      | 
 \    1 - 5*x  + 2*x/ 
----------------------
           2          
    1 - 5*x  + 2*x

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

Simplify

The third derivative [src]

              /                 2 \
              |     4*(-1 + 5*x)  |
-2*(-1 + 5*x)*|15 + --------------|
              |            2      |
              \     1 - 5*x  + 2*x/
-----------------------------------
                         2         
         /       2      \          
         \1 - 5*x  + 2*x/

- \frac{2 \left(5 x - 1\right) \left(\frac{4 \left(5 x - 1\right)^{2}}{- 5 x^{2} + 2 x + 1} + 15\right)}{\left(- 5 x^{2} + 2 x + 1\right)^{2}}

Simplify

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Derivative of y=lnsqrt(2x-5x^2+1)

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The solution