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2 answers:

7 0

1 1/2 + 1 1/3 is 2 5/6

maria [59]3 years ago

3 0

Answer:

Step-by-step explanation:

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NeX [460]

Answer:

$\displaystyle \frac{dy}{dx} \bigg| \limit_{(1, 4)} = 2$

General Formulas and Concepts:

<u>Pre-Algebra</u>

Order of Operations: BPEMDAS

Equality Properties

<u>Algebra I</u>

<u>Calculus</u>

Derivatives

Derivative Notation

Derivative of a constant is 0

Implicit Differentiation

Basic Power Rule:

Step-by-step explanation:

<u>Step 1: Define</u>

$\displaystyle \sqrt{x} - \sqrt{y} = -1$

Point (1, 4)

<u>Step 2: Differentiate</u>

[Function] Rewrite [Exponential Rule - Root Rewrite]: $\displaystyle x^{\frac{1}{2}} - y^{\frac{1}{2}} = -1$
[Implicit Differentiation] Basic Power Rule: $\displaystyle \frac{1}{2}x^{\frac{1}{2} - 1} - \frac{1}{2}y^{\frac{1}{2} - 1}\frac{dy}{dx} = 0$
[Implicit Differentiation] Simplify Exponents: $\displaystyle \frac{1}{2}x^{\frac{-1}{2}} - \frac{1}{2}y^{\frac{-1}{2}}\frac{dy}{dx} = 0$
[Implicit Differentiation] Rewrite [Exponential Rule - Rewrite]: $\displaystyle \frac{1}{2x^{\frac{1}{2}}} - \frac{1}{2y^{\frac{1}{2}}}\frac{dy}{dx} = 0$
[Implicit Differentiation] Isolate <em>y</em> terms: $\displaystyle -\frac{1}{2y^{\frac{1}{2}}}\frac{dy}{dx} = -\frac{1}{2x^{\frac{1}{2}}}$
[Implicit Differentiation] Isolate $\displaystyle \frac{dy}{dx}$ : $\displaystyle \frac{dy}{dx} = \frac{2y^{\frac{1}{2}}}{2x^{\frac{1}{2}}}$
[Implicit Differentiation] Simplify: $\displaystyle \frac{dy}{dx} = \frac{y^{\frac{1}{2}}}{x^{\frac{1}{2}}}$

<u>Step 3: Evaluate</u>

Substitute in point [Derivative]: $\displaystyle \frac{dy}{dx} = \frac{(4)^{\frac{1}{2}}}{(1)^{\frac{1}{2}}}$
Exponents: $\displaystyle \frac{dy}{dx} = \frac{2}{1}$
Division: $\displaystyle \frac{dy}{dx} = 2$