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# Find the implicit derivative $\frac{d}{dx}\left(\sin\left(x\right)=x\left(1+\tan\left(y\right)\right)\right)$

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##  Final answer to the problem

$y^{\prime}=\frac{-1-\tan\left(y\right)+\cos\left(x\right)}{x\sec\left(y\right)^2}$
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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
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Apply implicit differentiation by taking the derivative of both sides of the equation with respect to the differentiation variable

$\frac{d}{dx}\left(\sin\left(x\right)\right)=\frac{d}{dx}\left(x\left(1+\tan\left(y\right)\right)\right)$

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

$\frac{d}{dx}\left(\sin\left(x\right)\right)=\frac{d}{dx}\left(x\left(1+\tan\left(y\right)\right)\right)$

Learn how to solve implicit differentiation problems step by step online. Find the implicit derivative d/dx(sin(x)=x(1+tan(y))). Apply implicit differentiation by taking the derivative of both sides of the equation with respect to the differentiation variable. Apply the product rule for differentiation: (f\cdot g)'=f'\cdot g+f\cdot g', where f=. The derivative of the linear function is equal to 1. The derivative of the sine of a function is equal to the cosine of that function times the derivative of that function, in other words, if {f(x) = \sin(x)}, then {f'(x) = \cos(x)\cdot D_x(x)}.

##  Final answer to the problem

$y^{\prime}=\frac{-1-\tan\left(y\right)+\cos\left(x\right)}{x\sec\left(y\right)^2}$

##  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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a
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x
y
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(◻)
+
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×
◻/◻
/
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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

###  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.