Write an algebraic statement for the expression shown below.

If -y-2x^3=Y^2 then find D^2y/dx^2 at the point (-1,-2) in simplest form

algol13

Answer:

$\frac{d^2y}{dx^2} = \frac{-4}{3}$

General Formulas and Concepts:

Pre-Algebra

Order of Operations: BPEMDAS

Brackets
Parenthesis
Exponents
Multiplication
Division
Addition
Subtraction

Left to Right

Algebra I

Factoring

Calculus

Implicit Differentiation

The derivative of a constant is equal to 0

Basic Power Rule:

f(x) = cxⁿ
f’(x) = c·nxⁿ⁻¹

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

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

Quotient Rule: $\frac{d}{dx} [\frac{f(x)}{g(x)} ]=\frac{g(x)f'(x)-g'(x)f(x)}{g^2(x)}$

Step-by-step explanation:

Step 1: Define

-y - 2x³ = y²

Rate of change of tangent line at point (-1, -2)

Step 2: Differentiate Pt. 1

Find 1st Derivative

Implicit Differentiation [Basic Power Rule]: $-y'-6x^2=2yy'$
[Algebra] Isolate y' terms: $-6x^2=2yy'+y'$
[Algebra] Factor y': $-6x^2=y'(2y+1)$
[Algebra] Isolate y': $\frac{-6x^2}{(2y+1)}=y'$
[Algebra] Rewrite: $y' = \frac{-6x^2}{(2y+1)}$

Step 3: Differentiate Pt. 2

Find 2nd Derivative

Differentiate [Quotient Rule/Basic Power Rule]: $y'' = \frac{-12x(2y+1)+6x^2(2y')}{(2y+1)^2}$
[Derivative] Simplify: $y'' = \frac{-24xy-12x+12x^2y'}{(2y+1)^2}$
[Derivative] Back-Substitute y': $y'' = \frac{-24xy-12x+12x^2(\frac{-6x^2}{2y+1} )}{(2y+1)^2}$
[Derivative] Simplify: $y'' = \frac{-24xy-12x-\frac{72x^4}{2y+1} }{(2y+1)^2}$

Step 4: Find Slope at Given Point

[Algebra] Substitute in x and y: $y''(-1,-2) = \frac{-24(-1)(-2)-12(-1)-\frac{72(-1)^4}{2(-2)+1} }{(2(-2)+1)^2}$
[Pre-Algebra] Exponents: $y''(-1,-2) = \frac{-24(-1)(-2)-12(-1)-\frac{72(1)}{2(-2)+1} }{(2(-2)+1)^2}$
[Pre-Algebra] Multiply: $y''(-1,-2) = \frac{-48+12-\frac{72}{-4+1} }{(-4+1)^2}$
[Pre-Algebra] Add: $y''(-1,-2) = \frac{-36-\frac{72}{-3} }{(-3)^2}$
[Pre-Algebra] Exponents: $y''(-1,-2) = \frac{-36-\frac{72}{-3} }{9}$
[Pre-Algebra] Divide: $y''(-1,-2) = \frac{-36+24 }{9}$
[Pre-Algebra] Add: $y''(-1,-2) = \frac{-12}{9}$
[Pre-Algebra] Simplify: $y''(-1,-2) = \frac{-4}{3}$

6 0

3 years ago

logan drov from new orland to nashvill to visit friends his speed limit is 60 miles per hour if logans driving for the trip was

Alina [70]

Answer:

540 miles

Step-by-step explanation:

60 times 9

4 0

3 years ago

Why is the answer to a subtraction problem called the difference...

lana [24]

Maybe because you're finding the "difference" between the two numbers. We could also question why the answer to an addition problem is called sum.
Well, 'sum'body said that's what they should be called a long time ago, that's why we're using that term to call them now.

3 0

3 years ago

A right angle triangle is shown with hypotenuse equal to 10 centimeters. An acute angle of the triangle is labeled as x degrees

Nostrana [21]

The Question is incomplete the Complete Question is

Look at the triangle: A right angle triangle is shown with hypotenuse equal to 10 centimeters. An acute angle of the triangle is labeled as x degrees. The side adjacent to the acute angle has length 6 centimeters and the side opposite to the acute angle has length 8 centimeters. What is the value of tan x°?

Answer:

Therefore the value of tan x is

$\tan x=\dfrac{4}{3}$

Step-by-step explanation:

Given:

hypotenuse = 10 cm'

side adjacent to the acute angle 'x' = 6 cm.

side opposite to the acute angle 'x' = 8 cm.

To Find:

tan x = ?

Solution:

In Right Angle Triangle , Tan Identity we have

$\tan x= \dfrac{\textrm{side opposite to angle x}}{\textrm{side adjacent to angle x}}$

Substituting the values we get

$\tan x= \dfrac{8}{6}=\dfrac{4}{3}$

Therefore the value of tan x is

$\tan x=\dfrac{4}{3}$

4 0

3 years ago

BRAINLEST what is the value of the function at x=2

BARSIC [14]

Answer:

y = - 3

Step-by-step explanation:

Locate x = 2 on the x- axis, then go down vertically to meet the line. At this point read off the corresponding value on the y- axis, thus

The value of the function at x = 2 is y = - 3

3 0

3 years ago