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# Find the implicit derivative $\frac{d}{dx}\left(-x^2+\ln\left(1+xy^2\right)+\sqrt{x^2+y}\right)=y^{5x^3}$

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##  Final answer to the problem

$-2x+\frac{1}{1+xy^2}\left(y^2+2xy\cdot y^{\prime}\right)+\frac{1}{2\sqrt{x^2+y}}\left(2x+y^{\prime}\right)=y^{5x^3}$
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##  Step-by-step Solution 

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• Find the derivative using the definition
• Exact Differential Equation
• Linear Differential Equation
• Separable Differential Equation
• Homogeneous Differential Equation
• Find the derivative using the product rule
• Find the derivative using the quotient rule
• Find the derivative using logarithmic differentiation
• Find the derivative
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The derivative of a sum of two or more functions is the sum of the derivatives of each function

$\frac{d}{dx}\left(-x^2\right)+\frac{d}{dx}\left(\ln\left(1+xy^2\right)\right)+\frac{d}{dx}\left(\sqrt{x^2+y}\right)=y^{5x^3}$

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

$\frac{d}{dx}\left(-x^2\right)+\frac{d}{dx}\left(\ln\left(1+xy^2\right)\right)+\frac{d}{dx}\left(\sqrt{x^2+y}\right)=y^{5x^3}$

Learn how to solve implicit differentiation problems step by step online. Find the implicit derivative d/dx(-x^2+ln(1+xy^2)(x^2+y)^(1/2))=y^(5x^3). The derivative of a sum of two or more functions is the sum of the derivatives of each function. The derivative of a function multiplied by a constant is equal to the constant times the derivative of the function. The power rule for differentiation states that if n is a real number and f(x) = x^n, then f'(x) = nx^{n-1}. The power rule for differentiation states that if n is a real number and f(x) = x^n, then f'(x) = nx^{n-1}.

##  Final answer to the problem

$-2x+\frac{1}{1+xy^2}\left(y^2+2xy\cdot y^{\prime}\right)+\frac{1}{2\sqrt{x^2+y}}\left(2x+y^{\prime}\right)=y^{5x^3}$

##  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.