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(two x^ four -3sinx)*log(x,2)
(2x to the power of 4 minus 3 sinus of x) multiply by logarithm of (x,2)
(two x to the power of four minus 3 sinus of x) multiply by logarithm of (x,2)
(2x4-3sinx)*log(x,2)
2x4-3sinx*logx,2
(2x⁴-3sinx)*log(x,2)
(2x^4-3sinx)log(x,2)
(2x4-3sinx)log(x,2)
2x4-3sinxlogx,2
2x^4-3sinxlogx,2
Similar expressions
(2x^4+3sinx)*log(x,2)

The derivative of the function/ (2x^4-3sinx)*log(x,2)

Derivative of (2x^4-3sinx)*log(x,2)

The solution

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/   4           \ log(x)
\2*x  - 3*sin(x)/*------
                  log(2)

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

(2*x^4 - 3*sin(x))*(log(x)/log(2))

Detail solution

Apply the quotient rule, which is:

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

$f{\left(x \right)} = \left(2 x^{4} - 3 \sin{\left(x \right)}\right) \log{\left(x \right)}$ and $g{\left(x \right)} = \log{\left(2 \right)}$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Apply the product rule:
  
  $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
  
  $f{\left(x \right)} = 2 x^{4} - 3 \sin{\left(x \right)}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
  1. Differentiate $2 x^{4} - 3 \sin{\left(x \right)}$ term by term:
    1. The derivative of a constant times a function is the constant times the derivative of the function.
      1. The derivative of sine is cosine:
        
        $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
      So, the result is: $- 3 \cos{\left(x \right)}$
    2. The derivative of a constant times a function is the constant times the derivative of the function.
      1. Apply the power rule: $x^{4}$ goes to $4 x^{3}$
      So, the result is: $8 x^{3}$
    The result is: $8 x^{3} - 3 \cos{\left(x \right)}$
  $g{\left(x \right)} = \log{\left(x \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
  1. The derivative of $\log{\left(x \right)}$ is $\frac{1}{x}$ .
  The result is: $\left(8 x^{3} - 3 \cos{\left(x \right)}\right) \log{\left(x \right)} + \frac{2 x^{4} - 3 \sin{\left(x \right)}}{x}$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. The derivative of the constant $\log{\left(2 \right)}$ is zero.
Now plug in to the quotient rule:

$\frac{\left(8 x^{3} - 3 \cos{\left(x \right)}\right) \log{\left(x \right)} + \frac{2 x^{4} - 3 \sin{\left(x \right)}}{x}}{\log{\left(2 \right)}}$
Now simplify:

$\frac{2 x^{4} + x \left(8 x^{3} - 3 \cos{\left(x \right)}\right) \log{\left(x \right)} - 3 \sin{\left(x \right)}}{x \log{\left(2 \right)}}$

The answer is:

$\frac{2 x^{4} + x \left(8 x^{3} - 3 \cos{\left(x \right)}\right) \log{\left(x \right)} - 3 \sin{\left(x \right)}}{x \log{\left(2 \right)}}$

The graph

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Plot the graph f'(x)

The first derivative [src]

   4              /               3\       
2*x  - 3*sin(x)   \-3*cos(x) + 8*x /*log(x)
--------------- + -------------------------
    x*log(2)                log(2)

$$\frac{\left(8 x^{3} - 3 \cos{\left(x \right)}\right) \log{\left(x \right)}}{\log{\left(2 \right)}} + \frac{2 x^{4} - 3 \sin{\left(x \right)}}{x \log{\left(2 \right)}}$$

Simplify

The second derivative [src]

                 4     /               3\                           
  -3*sin(x) + 2*x    2*\-3*cos(x) + 8*x /     /   2         \       
- ---------------- + -------------------- + 3*\8*x  + sin(x)/*log(x)
          2                   x                                     
         x                                                          
--------------------------------------------------------------------
                               log(2)

$$\frac{3 \left(8 x^{2} + \sin{\left(x \right)}\right) \log{\left(x \right)} + \frac{2 \left(8 x^{3} - 3 \cos{\left(x \right)}\right)}{x} - \frac{2 x^{4} - 3 \sin{\left(x \right)}}{x^{2}}}{\log{\left(2 \right)}}$$

Simplify

The third derivative [src]

    /               3\     /               4\                                /   2         \
  3*\-3*cos(x) + 8*x /   2*\-3*sin(x) + 2*x /                              9*\8*x  + sin(x)/
- -------------------- + -------------------- + 3*(16*x + cos(x))*log(x) + -----------------
            2                      3                                               x        
           x                      x                                                         
--------------------------------------------------------------------------------------------
                                           log(2)

$$\frac{3 \left(16 x + \cos{\left(x \right)}\right) \log{\left(x \right)} + \frac{9 \left(8 x^{2} + \sin{\left(x \right)}\right)}{x} - \frac{3 \left(8 x^{3} - 3 \cos{\left(x \right)}\right)}{x^{2}} + \frac{2 \left(2 x^{4} - 3 \sin{\left(x \right)}\right)}{x^{3}}}{\log{\left(2 \right)}}$$

Simplify

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Derivative of (2x^4-3sinx)*log(x,2)

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The solution