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Sum of series:
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Identical expressions
f(x)=log^ zero ,5x- three
f(x) equally logarithm of to the power of 0,5x minus 3
f(x) equally logarithm of to the power of zero ,5x minus three
f(x)=log0,5x-3
fx=log0,5x-3
fx=log^0,5x-3
Similar expressions
f(x)=log^0,5x+3

The derivative of the function/ f(x)=log^0,5x-3

Derivative of f(x)=log^0,5x-3

The solution

You have entered [src]

  ____________
\/ log(x - 3)

$$\sqrt{\log{\left(x - 3 \right)}}$$

sqrt(log(x - 3))

Detail solution

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

$\frac{1}{2 \left(x - 3\right) \sqrt{\log{\left(x - 3 \right)}}}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

           1            
------------------------
            ____________
2*(x - 3)*\/ log(x - 3)

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

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Derivative of f(x)=log^0,5x-3

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The solution