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The derivative of the function/ f(x)=10logx/x

Derivative of f(x)=10logx/x

The solution

You have entered [src]

10*log(x)
---------
    x

$$\frac{10 \log{\left(x \right)}}{x}$$

(10*log(x))/x

Detail solution

Apply the quotient rule, which is:

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

$f{\left(x \right)} = 10 \log{\left(x \right)}$ 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. The derivative of $\log{\left(x \right)}$ is $\frac{1}{x}$ .
  So, the result is: $\frac{10}{x}$
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:

$\frac{10 - 10 \log{\left(x \right)}}{x^{2}}$
Now simplify:

$\frac{10 \left(1 - \log{\left(x \right)}\right)}{x^{2}}$

The answer is:

$\frac{10 \left(1 - \log{\left(x \right)}\right)}{x^{2}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

10   10*log(x)
-- - ---------
 2        2   
x        x

$$- \frac{10 \log{\left(x \right)}}{x^{2}} + \frac{10}{x^{2}}$$

Simplify

The second derivative [src]

10*(-3 + 2*log(x))
------------------
         3        
        x

$$\frac{10 \left(2 \log{\left(x \right)} - 3\right)}{x^{3}}$$

Simplify

The third derivative [src]

10*(11 - 6*log(x))
------------------
         4        
        x

$$\frac{10 \left(11 - 6 \log{\left(x \right)}\right)}{x^{4}}$$

Simplify