Derivative of ln3x^4

You have entered [src]

   4     
log (3*x)

$$\log{\left(3 x \right)}^{4}$$

log(3*x)^4

Detail solution

Let $u = \log{\left(3 x \right)}$ .
Apply the power rule: $u^{4}$ goes to $4 u^{3}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \log{\left(3 x \right)}$ :
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:
  
  $\frac{1}{x}$
The result of the chain rule is:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

  /                      2     \         
4*\6 - 9*log(3*x) + 2*log (3*x)/*log(3*x)
-----------------------------------------
                     3                   
                    x

$$\frac{4 \left(2 \log{\left(3 x \right)}^{2} - 9 \log{\left(3 x \right)} + 6\right) \log{\left(3 x \right)}}{x^{3}}$$

Simplify

The graph

Other calculators

How to use it?

Identical expressions

Similar expressions

Derivative of ln3x^4

The graph:

Piecewise:

The solution