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Find the implicit derivative $\frac{d}{dx}\left(e^{xy}=\arcsin\left(\mathrm{tanh}\left(y\right)\right)\right)$

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Solution

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Final answer to the problem

$y^{\prime}=\frac{-ye^{xy}\sqrt{1-\mathrm{tanh}\left(y\right)^2}}{xe^{xy}\sqrt{1-\mathrm{tanh}\left(y\right)^2}-\mathrm{sech}\left(y\right)^2}$
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Step-by-step Solution

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  • 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
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1

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

Learn how to solve integrals of rational functions problems step by step online.

$\frac{d}{dx}\left(e^{xy}\right)=\frac{d}{dx}\left(\arcsin\left(\mathrm{tanh}\left(y\right)\right)\right)$

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Learn how to solve integrals of rational functions problems step by step online. Find the implicit derivative d/dx(e^(xy)=arcsin(tanh(y))). Apply implicit differentiation by taking the derivative of both sides of the equation with respect to the differentiation variable. Applying the derivative of the exponential function. Apply the product rule for differentiation: (f\cdot g)'=f'\cdot g+f\cdot g', where f=x and g=y. The derivative of the linear function is equal to 1.

Final answer to the problem

$y^{\prime}=\frac{-ye^{xy}\sqrt{1-\mathrm{tanh}\left(y\right)^2}}{xe^{xy}\sqrt{1-\mathrm{tanh}\left(y\right)^2}-\mathrm{sech}\left(y\right)^2}$

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Function Plot

Plotting: $y^{\prime}=\frac{-ye^{xy}\sqrt{1-\mathrm{tanh}\left(y\right)^2}}{xe^{xy}\sqrt{1-\mathrm{tanh}\left(y\right)^2}-\mathrm{sech}\left(y\right)^2}$

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

◻√◻
◻√
∞
e
π
ln
log
log◻
lim
d/dx
D□x
∫
∫◻
|◻|
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:

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Main Topic: Integrals of Rational Functions

Integrals of rational functions of the form R(x) = P(x)/Q(x).

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