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# Find the implicit derivative $\frac{d}{dx}\left(y^x=x^y\right)$

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##  Final answer to the problem

$\frac{xy^x\ln\left(y\right)}{1-x}=\frac{x^{\left(2y-1\right)}}{1-x^y\ln\left(x\right)}$
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##  Step-by-step Solution 

How should I solve this problem?

• Choose an option
• Find the derivative using the definition
• Find the derivative using the product rule
• Find the derivative using the quotient rule
• Find the derivative using logarithmic differentiation
• Find the derivative
• Integrate by partial fractions
• Product of Binomials with Common Term
• FOIL Method
• Integrate by substitution
Can't find a method? Tell us so we can add it.
1

Apply implicit differentiation by taking the derivative of both sides of the equation with respect to the differentiation variable

$\frac{d}{dx}\left(y^x\right)=\frac{d}{dx}\left(x^y\right)$

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$\frac{d}{dx}\left(y^x\right)=\frac{d}{dx}\left(x^y\right)$

Learn how to solve problems step by step online. Find the implicit derivative d/dx(y^x=x^y). Apply implicit differentiation by taking the derivative of both sides of the equation with respect to the differentiation variable. The derivative \frac{d}{dx}\left(y^x\right) results in \frac{y^x\ln\left(y^x\right)}{1-x}. The derivative \frac{d}{dx}\left(x^y\right) results in \frac{x^{\left(2y-1\right)}}{1-x^y\ln\left(x\right)}. Simplify the derivative.

##  Final answer to the problem

$\frac{xy^x\ln\left(y\right)}{1-x}=\frac{x^{\left(2y-1\right)}}{1-x^y\ln\left(x\right)}$

##  Explore different ways to solve this problem

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

SnapXam A2

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