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The derivative of the function/ 5log3(x)-4lgx+8lnx

Derivative of 5log3(x)-4lgx+8lnx

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

The graph:

from to

Piecewise:

The solution

You have entered [src] 
  log(x)                      
5*------ - 4*log(x) + 8*log(x)
  log(3)
(−4log⁡(x)+5log⁡(x)log⁡(3))+8log⁡(x)\left(- 4 \log{\left(x \right)} + 5 \frac{\log{\left(x \right)}}{\log{\left(3 \right)}}\right) + 8 \log{\left(x \right)} 
5*(log(x)/log(3)) - 4*log(x) + 8*log(x)
Detail solution

  1. Differentiate (−4log⁡(x)+5log⁡(x)log⁡(3))+8log⁡(x)\left(- 4 \log{\left(x \right)} + 5 \frac{\log{\left(x \right)}}{\log{\left(3 \right)}}\right) + 8 \log{\left(x \right)} term by term:

    1. Differentiate −4log⁡(x)+5log⁡(x)log⁡(3)- 4 \log{\left(x \right)} + 5 \frac{\log{\left(x \right)}}{\log{\left(3 \right)}} term by term:

      1. The derivative of a constant times a function is the constant times the derivative of the function.

        1. The derivative of a constant times a function is the constant times the derivative of the function.

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

          So, the result is: 1xlog⁡(3)\frac{1}{x \log{\left(3 \right)}}

        So, the result is: 5xlog⁡(3)\frac{5}{x \log{\left(3 \right)}}

      2. The derivative of a constant times a function is the constant times the derivative of the function.

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

        So, the result is: −4x- \frac{4}{x}

      The result is: −4x+5xlog⁡(3)- \frac{4}{x} + \frac{5}{x \log{\left(3 \right)}}

    2. The derivative of a constant times a function is the constant times the derivative of the function.

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

      So, the result is: 8x\frac{8}{x}

    The result is: 4x+5xlog⁡(3)\frac{4}{x} + \frac{5}{x \log{\left(3 \right)}}

  2. Now simplify:

    log⁡(81)+5xlog⁡(3)\frac{\log{\left(81 \right)} + 5}{x \log{\left(3 \right)}}

The answer is:

log⁡(81)+5xlog⁡(3)\frac{\log{\left(81 \right)} + 5}{x \log{\left(3 \right)}}

The graph
02468-8-6-4-2-1010-200200
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
4      5    
- + --------
x   x*log(3)
4x+5xlog⁡(3)\frac{4}{x} + \frac{5}{x \log{\left(3 \right)}}
Simplify
The second derivative [src] 
 /      5   \ 
-|4 + ------| 
 \    log(3)/ 
--------------
       2      
      x
−4+5log⁡(3)x2- \frac{4 + \frac{5}{\log{\left(3 \right)}}}{x^{2}}
Simplify
The third derivative [src] 
  /      5   \
2*|4 + ------|
  \    log(3)/
--------------
       3      
      x
2(4+5log⁡(3))x3\frac{2 \left(4 + \frac{5}{\log{\left(3 \right)}}\right)}{x^{3}}
Simplify
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