Step-by-step Solution

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Step-by-step explanation

Problem to solve:

$\frac{d}{dx}\left(x\sqrt{1+y}+\sqrt{y\cdot\left(1+2x\right)}=2x\right)$

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

$\frac{d}{dx}\left(x\sqrt{1+y}+\sqrt{y\left(1+2x\right)}\right)=\frac{d}{dx}\left(2x\right)$

Learn how to solve implicit differentiation problems step by step online. Find the implicit derivative (d/dx)(x(1+y)^0.5+(y(1+2x))^0.5=2x). Apply implicit differentiation by taking the derivative of both sides of the equation with respect to the differentiation variable. The power of a product is equal to the product of it's factors raised to the same power. The derivative of a function multiplied by a constant (2) is equal to the constant times the derivative of the function. The derivative of the linear function is equal to 1.

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

Problem Analysis

$\frac{d}{dx}\left(x\sqrt{1+y}+\sqrt{y\cdot\left(1+2x\right)}=2x\right)$

Main topic:

Implicit differentiation

~ 0.2 seconds