Implicit Differentiation Calculator

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Example

Created on 2024-06-20Asked by Avery Rodriguez (Solvelet student)

Find

\frac{dy}{dx}

by implicit differentiation for the equation

x^2 + y^2 = 25

Solution

To find

\frac{dy}{dx}

by implicit differentiation for the equation

x^2 + y^2 = 25

: Differentiate both sides of the equation with respect to

x

\frac{d}{dx}(x^2 + y^2) = \frac{d}{dx}(25)

Using the chain rule:

\frac{d}{dx}(x^2) + \frac{d}{dx}(y^2) = 0

2x + 2y \frac{dy}{dx} = 0

Solve for

\frac{dy}{dx}

2y \frac{dy}{dx} = -2x

\frac{dy}{dx} = -\frac{x}{y}

Therefore, the derivative

\frac{dy}{dx}

for the equation

x^2 + y^2 = 25

is:

\frac{dy}{dx} = -\frac{x}{y}

Solved on Solvelet with Basic AI Model

Some of the related questions asked by Scarlett Lopez on Solvelet

1. Find the derivative of the implicitly defined function

x^2 + y^2 = 25

with respect to

x

,2. Determine whether the improper integral

\int_{0}^{\infty} e^{-x} \, dx

converges, and if it does, find its value.,

DefinitionImplicit differentiation is a way to do differentiate implicitly defined functions. When using explicit differentiation isolates the function on one side, and implicit differentiation differentiates both sides of an equation with respect to the independant variable, using the chain rule as required. It also helps to solve Equations that is quite difficult to solve writing one variable in terms of another. Examples: For the equation x2 + y2 = 1, differentiating both sides by x, 2x + 2ydxdy = 0 and thus dxdy = −y/x.

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