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

Step-by-step Solution

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Final Answer

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

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

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

Learn how to solve implicit differentiation problems step by step online.

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

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Learn how to solve implicit differentiation 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)}{-x+1}. The derivative \frac{d}{dx}\left(x^y\right) results in \frac{\left(x^y\right)^2}{\left(1-x^y\ln\left(x\right)\right)x}.

Final Answer

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

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1
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5
6
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

How to improve your answer:

Main Topic: Implicit Differentiation

Implicit differentiation makes use of the chain rule to differentiate implicitly defined functions. For differentiating an implicit function y(x), defined by an equation R(x, y) = 0, it is not generally possible to solve it explicitly for y(x) and then differentiate. Instead, one can differentiate R(x, y) with respect to x and y and then solve a linear equation in dy/dx for getting explicitly the derivative in terms of x and y.

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