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Mathematics, 07.06.2020 18:57 luckylady

Help! Stuck on this question (calculus) If $y=\frac{3}{x^2}$

work out $\frac{dx}{dy}$

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Answer from: delincuent5641

$\displaystyle \frac{dx}{dy} = \frac{3\sqrt{\frac{3}{y}}(y - 1)}{6y}$

General Formulas and Concepts:

Algebra I

Terms/CoefficientsFunctionsFunction Notation

Calculus

Derivatives

Derivative Notation

Basic Power Rule:

f(x) = cxⁿ f’(x) = c·nxⁿ⁻¹

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

Implicit Differentiation

Step-by-step explanation:

Step 1: Define

Identify

$\displaystyle y = \frac{3}{x^2}$

Step 2: Differentiate

Rewrite Function: $\displaystyle yx^2 = 3$ [Implicit Differentiation] Basic Power Rule [Product Rule]: $\displaystyle y^{1 - 1}x^2 + y(2x^{2 - 1})\frac{dx}{dy} = 3$ Simplify: $\displaystyle x^2 + y(2x)\frac{dx}{dy} = 3$ Isolate $\displaystyle \frac{dx}{dy}$ term: $\displaystyle y(2x)\frac{dx}{dy} = 3 - x^2$ Isolate $\displaystyle \frac{dx}{dy}$ : $\displaystyle \frac{dx}{dy} = \frac{3 - x^2}{2xy}$ Substitute in x²: $\displaystyle \frac{dx}{dy} = \frac{3 - \frac{3}{y}}{2xy}$ Simplify: $\displaystyle \frac{dx}{dy} = \frac{3(y - 1)}{2xy^2}$ Substitute in x: $\displaystyle \frac{dx}{dy} = \frac{3(y - 1)}{2y^2(\sqrt{\frac{3}{y}})}$ Simplify: $\displaystyle \frac{dx}{dy} = \frac{3\sqrt{\frac{3}{y}}(y - 1)}{6y}$

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

Unit: Derivatives

Book: College Calculus 10e

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Answer from: Quest

this was already solved by me with a different person. if you look it up in the search bar at the top of the page, you will get the answer.

step-by-step explanation:

Answer this 19 points and brainliest!

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Answer from: Quest

true

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