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Derivative of 6x^8-6lnx+3log3(x)

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   8              3*log(x)
6*x  - 6*log(x) + --------
                   log(3)

$$6 x^{8} - 6 \log{\left(x \right)} + \frac{3 \log{\left(x \right)}}{\log{\left(3 \right)}}$$

d /   8              3*log(x)\
--|6*x  - 6*log(x) + --------|
dx\                   log(3) /

$$\frac{d}{d x} \left(6 x^{8} - 6 \log{\left(x \right)} + \frac{3 \log{\left(x \right)}}{\log{\left(3 \right)}}\right)$$

Transform

Detail solution

Differentiate $6 x^{8} - 6 \log{\left(x \right)} + \frac{3 \log{\left(x \right)}}{\log{\left(3 \right)}}$ term by term:
1. The derivative of a constant times a function is the constant times the derivative of the function.
  1. Apply the power rule: $x^{8}$ goes to $8 x^{7}$
  So, the result is: $48 x^{7}$
2. The derivative of a constant times a function is the constant times the derivative of the function.
  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: $\frac{6}{x}$
  So, the result is: $- \frac{6}{x}$
3. 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: $\frac{3}{x \log{\left(3 \right)}}$
The result is: $48 x^{7} - \frac{6}{x} + \frac{3}{x \log{\left(3 \right)}}$
Now simplify:

$\frac{3 \left(\log{\left(3^{16 x^{8} - 2} \right)} + 1\right)}{x \log{\left(3 \right)}}$

The answer is:

$\frac{3 \left(\log{\left(3^{16 x^{8} - 2} \right)} + 1\right)}{x \log{\left(3 \right)}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

  6       7      3    
- - + 48*x  + --------
  x           x*log(3)

$$48 x^{7} - \frac{6}{x} + \frac{3}{x \log{\left(3 \right)}}$$

Simplify

The second derivative [src]

  /2         6       1    \
3*|-- + 112*x  - ---------|
  | 2             2       |
  \x             x *log(3)/

$$3 \cdot \left(112 x^{6} - \frac{1}{x^{2} \log{\left(3 \right)}} + \frac{2}{x^{2}}\right)$$

Simplify

The third derivative [src]

  /  2         5       1    \
6*|- -- + 336*x  + ---------|
  |   3             3       |
  \  x             x *log(3)/

$$6 \cdot \left(336 x^{5} - \frac{2}{x^{3}} + \frac{1}{x^{3} \log{\left(3 \right)}}\right)$$

Simplify

The graph