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Derivative of:
Derivative of e^(-x^2)
Derivative of x^2/4
Derivative of -e^-x
Derivative of (x)^(1/2)
Identical expressions
y=x^ eight *ln(x)
y equally x to the power of 8 multiply by ln(x)
y equally x to the power of eight multiply by ln(x)
y=x8*ln(x)
y=x8*lnx
y=x⁸*ln(x)
y=x^8ln(x)
y=x8ln(x)
y=x8lnx
y=x^8lnx

The derivative of the function/ y=x^8*ln(x)

Derivative of y=x^8*ln(x)

The solution

You have entered [src]

 8       
x *log(x)

$$x^{8} \log{\left(x \right)}$$

x^8*log(x)

Detail solution

Apply the product rule:

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

$f{\left(x \right)} = x^{8}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
1. Apply the power rule: $x^{8}$ goes to $8 x^{7}$
$g{\left(x \right)} = \log{\left(x \right)}$ ; 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: $8 x^{7} \log{\left(x \right)} + x^{7}$
Now simplify:

$x^{7} \left(8 \log{\left(x \right)} + 1\right)$

The answer is:

$x^{7} \left(8 \log{\left(x \right)} + 1\right)$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

 7      7       
x  + 8*x *log(x)

$$8 x^{7} \log{\left(x \right)} + x^{7}$$

Simplify

The second derivative [src]

 6                 
x *(15 + 56*log(x))

$$x^{6} \left(56 \log{\left(x \right)} + 15\right)$$

Simplify

The third derivative [src]

   5                  
2*x *(73 + 168*log(x))

$$2 x^{5} \left(168 \log{\left(x \right)} + 73\right)$$

Simplify

The graph

Derivative of y=x^8*ln(x)