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Derivative of:
Derivative of 4/x^3
Derivative of (x+3)^2*(x+5)-1
Derivative of x^3+2
Derivative of x^(-2)
Limit of the function:
x^5*log(x)
Identical expressions
x^ five *log(x)
x to the power of 5 multiply by logarithm of (x)
x to the power of five multiply by logarithm of (x)
x5*log(x)
x5*logx
x⁵*log(x)
x^5log(x)
x5log(x)
x5logx
x^5logx

The derivative of the function/ x^5*log(x)

Derivative of x^5*log(x)

The solution

You have entered [src]

 5       
x *log(x)

$$x^{5} \log{\left(x \right)}$$

x^5*log(x)

Detail solution

Apply the product rule:

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

$f{\left(x \right)} = x^{5}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
1. Apply the power rule: $x^{5}$ goes to $5 x^{4}$
$g{\left(x \right)} = \log{\left(x \right)}$ ; 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: $5 x^{4} \log{\left(x \right)} + x^{4}$
Now simplify:

$x^{4} \left(5 \log{\left(x \right)} + 1\right)$

The answer is:

$x^{4} \left(5 \log{\left(x \right)} + 1\right)$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

 4      4       
x  + 5*x *log(x)

$$5 x^{4} \log{\left(x \right)} + x^{4}$$

Simplify

The second derivative [src]

 3                
x *(9 + 20*log(x))

$$x^{3} \left(20 \log{\left(x \right)} + 9\right)$$

Simplify

The third derivative [src]

 2                 
x *(47 + 60*log(x))

$$x^{2} \left(60 \log{\left(x \right)} + 47\right)$$

Simplify

The graph

Derivative of x^5*log(x)