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The derivative of the function/ log2(x)+3lnx+2lgx

Derivative of log2(x)+3lnx+2lgx

The solution

You have entered [src]

log(x)                      
------ + 3*log(x) + 2*log(x)
log(2)

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

log(x)/log(2) + 3*log(x) + 2*log(x)

Detail solution

Differentiate $\left(\frac{\log{\left(x \right)}}{\log{\left(2 \right)}} + 3 \log{\left(x \right)}\right) + 2 \log{\left(x \right)}$ term by term:
1. Differentiate $\frac{\log{\left(x \right)}}{\log{\left(2 \right)}} + 3 \log{\left(x \right)}$ 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 $\log{\left(x \right)}$ is $\frac{1}{x}$ .
    So, the result is: $\frac{1}{x \log{\left(2 \right)}}$
  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: $\frac{3}{x}$
  The result is: $\frac{1}{x \log{\left(2 \right)}} + \frac{3}{x}$
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: $\frac{2}{x}$
The result is: $\frac{1}{x \log{\left(2 \right)}} + \frac{5}{x}$
Now simplify:

$\frac{1 + \log{\left(32 \right)}}{x \log{\left(2 \right)}}$

The answer is:

$\frac{1 + \log{\left(32 \right)}}{x \log{\left(2 \right)}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

5      1    
- + --------
x   x*log(2)

$$\frac{1}{x \log{\left(2 \right)}} + \frac{5}{x}$$

Simplify

The second derivative [src]

 /      1   \ 
-|5 + ------| 
 \    log(2)/ 
--------------
       2      
      x

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

Simplify

The third derivative [src]

  /      1   \
2*|5 + ------|
  \    log(2)/
--------------
       3      
      x

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

Simplify