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The derivative of the function/ log2(x)+3log3x

Derivative of log2(x)+3log3x

The solution

You have entered [src]

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

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

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

Detail solution

Differentiate $\frac{\log{\left(x \right)}}{\log{\left(2 \right)}} + 3 \log{\left(3 x \right)}$ 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. Let $u = 3 x$ .
  2. The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
  3. Then, apply the chain rule. Multiply by $\frac{d}{d x} 3 x$ :
    1. The derivative of a constant times a function is the constant times the derivative of the function.
      1. Apply the power rule: $x$ goes to $1$
      So, the result is: $3$
    The result of the chain rule is:
    
    $\frac{1}{x}$
  So, the result is: $\frac{3}{x}$
The result is: $\frac{1}{x \log{\left(2 \right)}} + \frac{3}{x}$
Now simplify:

$\frac{1 + \log{\left(8 \right)}}{x \log{\left(2 \right)}}$

The answer is:

$\frac{1 + \log{\left(8 \right)}}{x \log{\left(2 \right)}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

3      1    
- + --------
x   x*log(2)

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify