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Find the derivative of $x\ln\left(x^x\right)$

Step-by-step Solution

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Final Answer

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

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

$\frac{d}{dx}\left(x\right)\ln\left(x^x\right)+x\frac{d}{dx}\left(\ln\left(x^x\right)\right)$

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$\frac{d}{dx}\left(x\right)\ln\left(x^x\right)+x\frac{d}{dx}\left(\ln\left(x^x\right)\right)$

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Learn how to solve definite integrals problems step by step online. Find the derivative of xln(x^x). Apply the product rule for differentiation: (f\cdot g)'=f'\cdot g+f\cdot g', where f=x and g=\ln\left(x^x\right). The derivative of the linear function is equal to 1. The derivative of the natural logarithm of a function is equal to the derivative of the function divided by that function. If f(x)=ln\:a (where a is a function of x), then \displaystyle f'(x)=\frac{a'}{a}. The derivative \frac{d}{dx}\left(x^x\right) results in \left(\ln\left(x\right)+1\right)x^x.

Final Answer

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

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Solving a math problem using different methods is important because it enhances understanding, encourages critical thinking, allows for multiple solutions, and develops problem-solving strategies. Read more

Find the derivativeFind derivative of xln(x^x) using the product ruleFind derivative of xln(x^x) using the quotient ruleFind derivative of xln(x^x) using logarithmic differentiationFind derivative of xln(x^x) using the definition

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

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7
8
9
0
a
b
c
d
f
g
m
n
u
v
w
x
y
z
.
(◻)
+
-
×
◻/◻
/
÷
2

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

asin
acos
atan
acot
asec
acsc

sinh
cosh
tanh
coth
sech
csch

asinh
acosh
atanh
acoth
asech
acsch

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Main Topic: Definite Integrals

Given a function f(x) and the interval [a,b], the definite integral is equal to the area that is bounded by the graph of f(x), the x-axis and the vertical lines x=a and x=b

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