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Derivative of:
Derivative of 2x²
Derivative of x^2*sqrt(1-x^2)
Derivative of y=3x²+5x+4
Derivative of x^3*cot(x)
Identical expressions
two *ln(3x)*x^- one
2 multiply by ln(3x) multiply by x to the power of minus 1
two multiply by ln(3x) multiply by x to the power of minus one
2*ln(3x)*x-1
2*ln3x*x-1
2ln(3x)x^-1
2ln(3x)x-1
2ln3xx-1
2ln3xx^-1
Similar expressions
2*ln(3x)*x^+1

The derivative of the function/ 2*ln(3x)*x^-1

Derivative of 2ln(3x)x^-1

The solution

You have entered [src]

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

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

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

Detail solution

Apply the quotient rule, which is:

$\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$

$f{\left(x \right)} = 2 \log{\left(3 x \right)}$ and $g{\left(x \right)} = x$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. 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: $\frac{2}{x}$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Apply the power rule: $x$ goes to $1$
Now plug in to the quotient rule:

$\frac{2 - 2 \log{\left(3 x \right)}}{x^{2}}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

2*(11 - 6*log(3*x))
-------------------
          4        
         x

$$\frac{2 \left(11 - 6 \log{\left(3 x \right)}\right)}{x^{4}}$$

Simplify