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Similar expressions
lnx*((x+1)^(1/2))

The derivative of the function/ lnx*((x-1)^(1/2))

Derivative of lnx*((x-1)^(1/2))

The solution

You have entered [src]

         _______
log(x)*\/ x - 1

$$\sqrt{x - 1} \log{\left(x \right)}$$

log(x)*sqrt(x - 1)

Detail solution

Apply the product rule:

$\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$

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

$\frac{\frac{x \log{\left(x \right)}}{2} + x - 1}{x \sqrt{x - 1}}$

The answer is:

$\frac{\frac{x \log{\left(x \right)}}{2} + x - 1}{x \sqrt{x - 1}}$

The graph

Plot the graph f(x)

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The first derivative [src]

  _______              
\/ x - 1       log(x)  
--------- + -----------
    x           _______
            2*\/ x - 1

$$\frac{\log{\left(x \right)}}{2 \sqrt{x - 1}} + \frac{\sqrt{x - 1}}{x}$$

Simplify

The second derivative [src]

                 ________                
     1         \/ -1 + x        log(x)   
------------ - ---------- - -------------
    ________        2                 3/2
x*\/ -1 + x        x        4*(-1 + x)

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

Simplify

The third derivative [src]

    ________                                                    
2*\/ -1 + x           3                 3             3*log(x)  
------------ - --------------- - --------------- + -------------
      3           2   ________               3/2             5/2
     x         2*x *\/ -1 + x    4*x*(-1 + x)      8*(-1 + x)

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

Simplify

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Derivative of lnx*((x-1)^(1/2))

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The solution