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Derivative of:
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Derivative of ln(4x)
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Derivative of 5-3x
Identical expressions
(log(x)/log(three))*ln^4x
( logarithm of (x) divide by logarithm of (3)) multiply by ln to the power of 4x
( logarithm of (x) divide by logarithm of (three)) multiply by ln to the power of 4x
(log(x)/log(3))*ln4x
logx/log3*ln4x
(log(x)/log(3))*ln⁴x
(log(x)/log(3))ln^4x
(log(x)/log(3))ln4x
logx/log3ln4x
logx/log3ln^4x
(log(x) divide by log(3))*ln^4x

The derivative of the function/ (log(x)/log(3))*ln^4x

Derivative of (log(x)/log(3))*ln^4x

The solution

You have entered [src]

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

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

(log(x)/log(3))*log(x)^4

Detail solution

Apply the quotient rule, which is:

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

$f{\left(x \right)} = \log{\left(x \right)}^{5}$ and $g{\left(x \right)} = \log{\left(3 \right)}$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Let $u = \log{\left(x \right)}$ .
2. Apply the power rule: $u^{5}$ goes to $5 u^{4}$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \log{\left(x \right)}$ :
  1. The derivative of $\log{\left(x \right)}$ is $\frac{1}{x}$ .
  The result of the chain rule is:
  
  $\frac{5 \log{\left(x \right)}^{4}}{x}$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. The derivative of the constant $\log{\left(3 \right)}$ is zero.
Now plug in to the quotient rule:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

   3                   
log (x)*(20 - 5*log(x))
-----------------------
        2              
       x *log(3)

$$\frac{\left(20 - 5 \log{\left(x \right)}\right) \log{\left(x \right)}^{3}}{x^{2} \log{\left(3 \right)}}$$

Simplify

The third derivative [src]

     2    /                      2   \
2*log (x)*\30 - 30*log(x) + 5*log (x)/
--------------------------------------
               3                      
              x *log(3)

$$\frac{2 \left(5 \log{\left(x \right)}^{2} - 30 \log{\left(x \right)} + 30\right) \log{\left(x \right)}^{2}}{x^{3} \log{\left(3 \right)}}$$

Simplify

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Identical expressions

Derivative of (log(x)/log(3))*ln^4x

The graph:

Piecewise:

The solution