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The derivative of the function/ y=ln5*log5(x)

Derivative of y=ln5*log5(x)

The solution

You have entered [src]

log(5)*log(x)
-------------
    log(5)

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

d /log(5)*log(x)\
--|-------------|
dx\    log(5)   /

$$\frac{d}{d x} \frac{\log{\left(5 \right)} \log{\left(x \right)}}{\log{\left(5 \right)}}$$

Transform

Detail solution

Apply the quotient rule, which is:

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

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

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. The derivative of $\log{\left(x \right)}$ is $\frac{1}{x}$ .
  So, the result is: $\frac{\log{\left(5 \right)}}{x}$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. The derivative of the constant $\log{\left(5 \right)}$ is zero.
Now plug in to the quotient rule:

$\frac{1}{x}$

The answer is:

$\frac{1}{x}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

1
-
x

$$\frac{1}{x}$$

Simplify

The second derivative [src]

-1 
---
  2
 x

$$- \frac{1}{x^{2}}$$

Simplify

3-th derivative [src]

2 
--
 3
x

$$\frac{2}{x^{3}}$$

Simplify

The third derivative [src]

2 
--
 3
x

$$\frac{2}{x^{3}}$$

Simplify

The graph

Derivative of y=ln5*log5(x)