Mister Exam

Lang:

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

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

The solution

You have entered [src]

  log(x)                      
5*------ - 4*log(x) + 8*log(x)
  log(3)

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

5*(log(x)/log(3)) - 4*log(x) + 8*log(x)

Detail solution

Differentiate $\left(- 4 \log{\left(x \right)} + 5 \frac{\log{\left(x \right)}}{\log{\left(3 \right)}}\right) + 8 \log{\left(x \right)}$ term by term:
1. Differentiate $- 4 \log{\left(x \right)} + 5 \frac{\log{\left(x \right)}}{\log{\left(3 \right)}}$ 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 $\log{\left(x \right)}$ is $\frac{1}{x}$ .
      So, the result is: $\frac{1}{x \log{\left(3 \right)}}$
    So, the result is: $\frac{5}{x \log{\left(3 \right)}}$
  2. The derivative of a constant times a function is the constant times the derivative of the function.
    1. The derivative of $\log{\left(x \right)}$ is $\frac{1}{x}$ .
    So, the result is: $- \frac{4}{x}$
  The result is: $- \frac{4}{x} + \frac{5}{x \log{\left(3 \right)}}$
2. The derivative of a constant times a function is the constant times the derivative of the function.
  1. The derivative of $\log{\left(x \right)}$ is $\frac{1}{x}$ .
  So, the result is: $\frac{8}{x}$
The result is: $\frac{4}{x} + \frac{5}{x \log{\left(3 \right)}}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

4      5    
- + --------
x   x*log(3)

$$\frac{4}{x} + \frac{5}{x \log{\left(3 \right)}}$$

Simplify

The second derivative [src]

 /      5   \ 
-|4 + ------| 
 \    log(3)/ 
--------------
       2      
      x

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

Simplify

The third derivative [src]

  /      5   \
2*|4 + ------|
  \    log(3)/
--------------
       3      
      x

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

Simplify