Mister Exam

Other calculators

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

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

Function f() - derivative -N order at the point
v

The graph:

from to

Piecewise:

The solution

You have entered [src] 
  ____________
\/ log(x - 3)
log⁡(x−3)\sqrt{\log{\left(x - 3 \right)}} 
sqrt(log(x - 3))
Detail solution

  1. Let u=log⁡(x−3)u = \log{\left(x - 3 \right)}.

  2. Apply the power rule: u\sqrt{u} goes to 12u\frac{1}{2 \sqrt{u}}

  3. Then, apply the chain rule. Multiply by ddxlog⁡(x−3)\frac{d}{d x} \log{\left(x - 3 \right)}:

    1. Let u=x−3u = x - 3.

    2. The derivative of log⁡(u)\log{\left(u \right)} is 1u\frac{1}{u}.

    3. Then, apply the chain rule. Multiply by ddx(x−3)\frac{d}{d x} \left(x - 3\right):

      1. Differentiate x−3x - 3 term by term:

        1. Apply the power rule: xx goes to 11

        2. The derivative of the constant −3-3 is zero.

        The result is: 11

      The result of the chain rule is:

      1x−3\frac{1}{x - 3}

    The result of the chain rule is:

    12(x−3)log⁡(x−3)\frac{1}{2 \left(x - 3\right) \sqrt{\log{\left(x - 3 \right)}}}

  4. Now simplify:

    12(x−3)log⁡(x−3)\frac{1}{2 \left(x - 3\right) \sqrt{\log{\left(x - 3 \right)}}}

The answer is:

12(x−3)log⁡(x−3)\frac{1}{2 \left(x - 3\right) \sqrt{\log{\left(x - 3 \right)}}}

The graph
3.00003.01003.00103.00203.00303.00403.00503.00603.00703.00803.00900.02-0.02
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
           1            
------------------------
            ____________
2*(x - 3)*\/ log(x - 3)
12(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)
−2+1log⁡(x−3)4(x−3)2log⁡(x−3)- \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)
1+34log⁡(x−3)+38log⁡(x−3)2(x−3)3log⁡(x−3)\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
    © Mister Exam – Calculator