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Identical expressions
y=x^ four (8ln^ two -x-4lnx+ one)
y equally x to the power of 4(8ln squared minus x minus 4lnx plus 1)
y equally x to the power of four (8ln to the power of two minus x minus 4lnx plus one)
y=x4(8ln2-x-4lnx+1)
y=x48ln2-x-4lnx+1
y=x⁴(8ln²-x-4lnx+1)
y=x to the power of 4(8ln to the power of 2-x-4lnx+1)
y=x^48ln^2-x-4lnx+1
Similar expressions
y=x^4(8ln^2-x+4lnx+1)
y=x^4(8ln^2-x-4lnx-1)
y=x^4(8ln^2+x-4lnx+1)

The derivative of the function/ y=x^4(8ln^2-x-4lnx+1)

Derivative of y=x^4(8ln^2-x-4lnx+1)

The solution

You have entered [src]

 4 /     2                      \
x *\8*log (x) - x - 4*log(x) + 1/

$$x^{4} \left(\left(\left(- x + 8 \log{\left(x \right)}^{2}\right) - 4 \log{\left(x \right)}\right) + 1\right)$$

x^4*(8*log(x)^2 - x - 4*log(x) + 1)

Detail solution

Apply the product rule:

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

$f{\left(x \right)} = x^{4}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
1. Apply the power rule: $x^{4}$ goes to $4 x^{3}$
$g{\left(x \right)} = \left(\left(- x + 8 \log{\left(x \right)}^{2}\right) - 4 \log{\left(x \right)}\right) + 1$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
1. Differentiate $\left(\left(- x + 8 \log{\left(x \right)}^{2}\right) - 4 \log{\left(x \right)}\right) + 1$ term by term:
  1. Differentiate $\left(- x + 8 \log{\left(x \right)}^{2}\right) - 4 \log{\left(x \right)}$ term by term:
    1. Differentiate $- x + 8 \log{\left(x \right)}^{2}$ term by term:
      1. The derivative of a constant times a function is the constant times the derivative of the function.
        
        Let $u = \log{\left(x \right)}$ .
        
        Apply the power rule: $u^{2}$ goes to $2 u$
        
        Then, apply the chain rule. Multiply by $\frac{d}{d x} \log{\left(x \right)}$ :
        
        The derivative of $\log{\left(x \right)}$ is $\frac{1}{x}$ .
        
        The result of the chain rule is:
        
        $\frac{2 \log{\left(x \right)}}{x}$
        
        So, the result is: $\frac{16 \log{\left(x \right)}}{x}$
      2. The derivative of a constant times a function is the constant times the derivative of the function.
        
        Apply the power rule: $x$ goes to $1$
        
        So, the result is: $-1$
      The result is: $-1 + \frac{16 \log{\left(x \right)}}{x}$
    2. 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: $- \frac{4}{x}$
    The result is: $-1 + \frac{16 \log{\left(x \right)}}{x} - \frac{4}{x}$
  2. The derivative of the constant $1$ is zero.
  The result is: $-1 + \frac{16 \log{\left(x \right)}}{x} - \frac{4}{x}$
The result is: $x^{4} \left(-1 + \frac{16 \log{\left(x \right)}}{x} - \frac{4}{x}\right) + 4 x^{3} \left(\left(\left(- x + 8 \log{\left(x \right)}^{2}\right) - 4 \log{\left(x \right)}\right) + 1\right)$
Now simplify:

$x^{3} \left(- 5 x + 32 \log{\left(x \right)}^{2}\right)$

The answer is:

$x^{3} \left(- 5 x + 32 \log{\left(x \right)}^{2}\right)$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

 4 /     4   16*log(x)\      3 /     2                      \
x *|-1 - - + ---------| + 4*x *\8*log (x) - x - 4*log(x) + 1/
   \     x       x    /

$$x^{4} \left(-1 + \frac{16 \log{\left(x \right)}}{x} - \frac{4}{x}\right) + 4 x^{3} \left(\left(\left(- x + 8 \log{\left(x \right)}^{2}\right) - 4 \log{\left(x \right)}\right) + 1\right)$$

Simplify

The second derivative [src]

    2 /           2                            /    4   16*log(x)\\
-4*x *|-8 - 24*log (x) + 3*x + 16*log(x) + 2*x*|1 + - - ---------||
      \                                        \    x       x    //

$$- 4 x^{2} \left(2 x \left(1 - \frac{16 \log{\left(x \right)}}{x} + \frac{4}{x}\right) + 3 x - 24 \log{\left(x \right)}^{2} + 16 \log{\left(x \right)} - 8\right)$$

Simplify

The third derivative [src]

    /                             2          /    4   16*log(x)\\
4*x*|52 - 64*log(x) - 6*x + 48*log (x) - 9*x*|1 + - - ---------||
    \                                        \    x       x    //

$$4 x \left(- 9 x \left(1 - \frac{16 \log{\left(x \right)}}{x} + \frac{4}{x}\right) - 6 x + 48 \log{\left(x \right)}^{2} - 64 \log{\left(x \right)} + 52\right)$$

Simplify

The graph

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Derivative of y=x^4(8ln^2-x-4lnx+1)

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The solution