# Step-by-step Solution

## Find the derivative using the product rule $\frac{d}{dx}\left(\sqrt{x}\ln\left(x\right)\right)$

Go!
1
2
3
4
5
6
7
8
9
0
x
y
(◻)
◻/◻
÷
2

e
π
ln
log
log
lim
d/dx
Dx
>
<
>=
<=
sin
cos
tan
cot
sec
csc

asin
acos
atan
acot
asec
acsc

sinh
cosh
tanh
coth
sech
csch

asinh
acosh
atanh
acoth
asech
acsch

### Videos

$\frac{1}{2}x^{-\frac{1}{2}}\ln\left(x\right)+x^{-\frac{1}{2}}$

## Step-by-step explanation

Problem to solve:

$\frac{d}{dx}\left(\sqrt{x}\ln\left(x\right)\right)$
1

Apply the product rule for differentiation: $(f\cdot g)'=f'\cdot g+f\cdot g'$, where $f=\sqrt{x}$ and $g=\ln\left(x\right)$

$\frac{d}{dx}\left(\sqrt{x}\right)\ln\left(x\right)+\sqrt{x}\cdot\frac{d}{dx}\left(\ln\left(x\right)\right)$
2

The power rule for differentiation states that if $n$ is a real number and $f(x) = x^n$, then $f'(x) = nx^{n-1}$

$\frac{1}{2}x^{-\frac{1}{2}}\ln\left(x\right)+\sqrt{x}\cdot\frac{d}{dx}\left(\ln\left(x\right)\right)$

$\frac{1}{2}x^{-\frac{1}{2}}\ln\left(x\right)+x^{-\frac{1}{2}}$
$\frac{d}{dx}\left(\sqrt{x}\ln\left(x\right)\right)$