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Identical expressions
y=(ln^ four)*(5x^ three -2x+ six)
y equally (ln to the power of 4) multiply by (5x cubed minus 2x plus 6)
y equally (ln to the power of four) multiply by (5x to the power of three minus 2x plus six)
y=(ln4)*(5x3-2x+6)
y=ln4*5x3-2x+6
y=(ln⁴)*(5x³-2x+6)
y=(ln to the power of 4)*(5x to the power of 3-2x+6)
y=(ln^4)(5x^3-2x+6)
y=(ln4)(5x3-2x+6)
y=ln45x3-2x+6
y=ln^45x^3-2x+6
Similar expressions
y=(ln^4)*(5x^3-2x-6)
y=(ln^4)*(5x^3+2x+6)

The derivative of the function/ y=(ln^4)*(5x^3-2x+6)

Derivative of y=(ln^4)*(5x^3-2x+6)

The solution

You have entered [src]

   4    /   3          \
log (x)*\5*x  - 2*x + 6/

$$\left(\left(5 x^{3} - 2 x\right) + 6\right) \log{\left(x \right)}^{4}$$

log(x)^4*(5*x^3 - 2*x + 6)

Detail solution

Apply the product rule:

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

$f{\left(x \right)} = \log{\left(x \right)}^{4}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
1. Let $u = \log{\left(x \right)}$ .
2. Apply the power rule: $u^{4}$ goes to $4 u^{3}$
3. 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:
  
  $\frac{4 \log{\left(x \right)}^{3}}{x}$
$g{\left(x \right)} = \left(5 x^{3} - 2 x\right) + 6$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
1. Differentiate $\left(5 x^{3} - 2 x\right) + 6$ term by term:
  1. Differentiate $5 x^{3} - 2 x$ 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: $x^{3}$ goes to $3 x^{2}$
      So, the result is: $15 x^{2}$
    2. The derivative of a constant times a function is the constant times the derivative of the function.
      1. Apply the power rule: $x$ goes to $1$
      So, the result is: $-2$
    The result is: $15 x^{2} - 2$
  2. The derivative of the constant $6$ is zero.
  The result is: $15 x^{2} - 2$
The result is: $\left(15 x^{2} - 2\right) \log{\left(x \right)}^{4} + \frac{4 \left(\left(5 x^{3} - 2 x\right) + 6\right) \log{\left(x \right)}^{3}}{x}$
Now simplify:

$\frac{\left(20 x^{3} + x \left(15 x^{2} - 2\right) \log{\left(x \right)} - 8 x + 24\right) \log{\left(x \right)}^{3}}{x}$

The answer is:

$\frac{\left(20 x^{3} + x \left(15 x^{2} - 2\right) \log{\left(x \right)} - 8 x + 24\right) \log{\left(x \right)}^{3}}{x}$

The graph

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Plot the graph f'(x)

The first derivative [src]

                            3    /   3          \
   4    /         2\   4*log (x)*\5*x  - 2*x + 6/
log (x)*\-2 + 15*x / + --------------------------
                                   x

$$\left(15 x^{2} - 2\right) \log{\left(x \right)}^{4} + \frac{4 \left(\left(5 x^{3} - 2 x\right) + 6\right) \log{\left(x \right)}^{3}}{x}$$

Simplify

The second derivative [src]

          /                               /             3\     /         2\       \
     2    |        2      2*(-3 + log(x))*\6 - 2*x + 5*x /   4*\-2 + 15*x /*log(x)|
2*log (x)*|15*x*log (x) - -------------------------------- + ---------------------|
          |                               2                            x          |
          \                              x                                        /

$$2 \left(15 x \log{\left(x \right)}^{2} + \frac{4 \left(15 x^{2} - 2\right) \log{\left(x \right)}}{x} - \frac{2 \left(\log{\left(x \right)} - 3\right) \left(5 x^{3} - 2 x + 6\right)}{x^{2}}\right) \log{\left(x \right)}^{2}$$

Simplify

The third derivative [src]

  /                             /                    2   \ /             3\                   /         2\       \       
  |      3             2      2*\6 - 9*log(x) + 2*log (x)/*\6 - 2*x + 5*x /   6*(-3 + log(x))*\-2 + 15*x /*log(x)|       
2*|15*log (x) + 180*log (x) + --------------------------------------------- - -----------------------------------|*log(x)
  |                                                  3                                          2                |       
  \                                                 x                                          x                 /

$$2 \left(15 \log{\left(x \right)}^{3} + 180 \log{\left(x \right)}^{2} - \frac{6 \left(15 x^{2} - 2\right) \left(\log{\left(x \right)} - 3\right) \log{\left(x \right)}}{x^{2}} + \frac{2 \left(5 x^{3} - 2 x + 6\right) \left(2 \log{\left(x \right)}^{2} - 9 \log{\left(x \right)} + 6\right)}{x^{3}}\right) \log{\left(x \right)}$$

Simplify

The graph

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Derivative of y=(ln^4)*(5x^3-2x+6)

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The solution