Mathematics, 19.07.2021 01:50, 20jmurphy82

Differentiate the following Functions
5x^2-2xy + 4y^3= 5

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answered: amandasantiago2001

$\displaystyle y' = \frac{y - 5x}{x + 6y^2}$

General Formulas and Concepts:

Algebra I

Terms/CoefficientsFactoring

Calculus

Differentiation

DerivativesDerivative NotationImplicit Differentiation

Derivative Property [Multiplied Constant]: $\displaystyle \frac{d}{dx} [cf(x)] = c \cdot f'(x)$

Derivative Property [Addition/Subtraction]: $\displaystyle \frac{d}{dx}[f(x) + g(x)] = \frac{d}{dx}[f(x)] + \frac{d}{dx}[g(x)]$

Basic Power Rule:

f(x) = cxⁿf’(x) = c·nxⁿ⁻¹

Derivative Rule [Product Rule]: $\displaystyle \frac{d}{dx} [f(x)g(x)]=f'(x)g(x) + g'(x)f(x)$

Derivative Rule [Chain Rule]: $\displaystyle \frac{d}{dx}[f(g(x))] =f'(g(x)) \cdot g'(x)$

Step-by-step explanation:

Step 1: Define

Identify

$\displaystyle 5x^2 - 2xy + 4y^3 = 5$

Step 2: Differentiate

Implicit Differentiation: $\displaystyle \frac{dy}{dx}[5x^2 - 2xy + 4y^3] = \frac{dy}{dx}[5]$ Rewrite [Derivative Property - Addition/Subtraction]: $\displaystyle \frac{dy}{dx}[5x^2] - \frac{dy}{dx}[2xy] + \frac{dy}{dx}[4y^3] = \frac{dy}{dx}[5]$ Rewrite [Derivative Property - Multiplied Constant]: $\displaystyle 5\frac{dy}{dx}[x^2] - 2\frac{dy}{dx}[xy] + 4\frac{dy}{dx}[y^3] = \frac{dy}{dx}[5]$ Basic Power Rule [Chain Rule]: $\displaystyle 10x - 2\frac{dy}{dx}[xy] + 12y^2y' = 0$ Product Rule: $\displaystyle 10x - 2\bigg[ \frac{dy}{dx}[x]y + x\frac{dy}{dx}[y] \bigg] + 12y^2y' = 0$ Basic Power Rule [Chain Rule]: $\displaystyle 10x - 2\bigg[ y + xy' \bigg] + 12y^2y' = 0$ Simplify: $\displaystyle 10x - 2y + 2xy' + 12y^2y' = 0$ Isolate y' terms: $\displaystyle 2xy' + 12y^2y' = 2y - 10x$ Factor: $\displaystyle y'(2x + 12y^2) = 2y - 10x$ Isolate y': $\displaystyle y' = \frac{2y - 10x}{2x + 12y^2}$ Factor: $\displaystyle y' = \frac{2(y - 5x)}{2(x + 6y^2)}$ Simplify: $\displaystyle y' = \frac{y - 5x}{x + 6y^2}$

Topic: AP Calculus AB/BC (Calculus I/I + II)

Unit: Differentiation

Book: College Calculus 10e

answered: Guest

sum= when you add is how much the whole amount is

distance= when you use angles and it means how far it goes

step-by-step explanation:

answered: Guest

is there more to this question?