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Derivative of:
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Derivative of 5^-x
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Identical expressions
y=x^ five *ln(x)
y equally x to the power of 5 multiply by ln(x)
y equally x to the power of five multiply by ln(x)
y=x5*ln(x)
y=x5*lnx
y=x⁵*ln(x)
y=x^5ln(x)
y=x5ln(x)
y=x5lnx
y=x^5lnx
Similar expressions
y=x^5*lnx

The derivative of the function/ y=x^5*ln(x)

Derivative of y=x^5*ln(x)

The solution

You have entered [src]

 5       
x *log(x)

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

d / 5       \
--\x *log(x)/
dx

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

Transform

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:

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

The answer is:

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

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))

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

Simplify

The third derivative [src]

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

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

Simplify

The graph

Derivative of y=x^5*ln(x)