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Identical expressions
y=x^6x+ one /lnx
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y equally x to the power of 6x plus one divide by lnx
y=x6x+1/lnx
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y=x^6x+1 divide by lnx
Similar expressions
y=x^6x-1/lnx

The derivative of the function/ y=x^6x+1/lnx

Derivative of y=x^6x+1/lnx

The solution

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 6       1   
x *x + ------
       log(x)

$$x x^{6} + \frac{1}{\log{\left(x \right)}}$$

x^6*x + 1/log(x)

Detail solution

Differentiate $x x^{6} + \frac{1}{\log{\left(x \right)}}$ term by term:
1. Apply the product rule:
  
  $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
  
  $f{\left(x \right)} = x^{6}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
  1. Apply the power rule: $x^{6}$ goes to $6 x^{5}$
  $g{\left(x \right)} = x$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
  1. Apply the power rule: $x$ goes to $1$
  The result is: $7 x^{6}$
2. Let $u = \log{\left(x \right)}$ .
3. Apply the power rule: $\frac{1}{u}$ goes to $- \frac{1}{u^{2}}$
4. 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:
  
  $- \frac{1}{x \log{\left(x \right)}^{2}}$
The result is: $7 x^{6} - \frac{1}{x \log{\left(x \right)}^{2}}$

The answer is:

$7 x^{6} - \frac{1}{x \log{\left(x \right)}^{2}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

   6       1    
7*x  - ---------
            2   
       x*log (x)

$$7 x^{6} - \frac{1}{x \log{\left(x \right)}^{2}}$$

Simplify

The second derivative [src]

    5       1            2     
42*x  + ---------- + ----------
         2    2       2    3   
        x *log (x)   x *log (x)

$$42 x^{5} + \frac{1}{x^{2} \log{\left(x \right)}^{2}} + \frac{2}{x^{2} \log{\left(x \right)}^{3}}$$

Simplify

The third derivative [src]

  /     4       1            3            3     \
2*|105*x  - ---------- - ---------- - ----------|
  |          3    2       3    4       3    3   |
  \         x *log (x)   x *log (x)   x *log (x)/

$$2 \left(105 x^{4} - \frac{1}{x^{3} \log{\left(x \right)}^{2}} - \frac{3}{x^{3} \log{\left(x \right)}^{3}} - \frac{3}{x^{3} \log{\left(x \right)}^{4}}\right)$$

Simplify

The graph

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Derivative of y=x^6x+1/lnx

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The solution