Mathematics High School

Use implicit differentiation to find zº+y³ = 10 dy = dr Question Help: Video Submit Question dy da without first solving for y. 0/1 pt 399 Details Details SLOWL n Question 2 Use implicit differentiation to find z² y² = 1 64 81 dy = dz At the given point, find the slope. dy da (3.8.34) Question Help: Video dy dz without first solving for y. 0/1 pt 399 Details Question 3 Use implicit differentiation to find 4 4x² + 3x + 2y <= 110 dy dz At the given point, find the slope. dy dz (-5.-5) Question Help: Video Submit Question || dy dz without first solving for y. 0/1 pt 399 Details Submit Question Question 4 B0/1 pt 399 Details Given the equation below, find 162 +1022y + y² = 27 dy dz Now, find the equation of the tangent line to the curve at (1, 1). Write your answer in mz + b format Y Question Help: Video Submit Question dy dz Question 5 Find the slope of the tangent line to the curve -2²-3ry-2y³ = -76 at the point (2, 3). Question Help: Video Submit Question Question 6 Find the slope of the tangent line to the curve (a lemniscate) 2(x² + y²)² = 25(x² - y²) at the point (3, -1) slope = Question Help: Video 0/1 pt 399 Details 0/1 pt 399 Details

Answers

Answer 1

The given problem can be solved separetely. Let's solve each of the given problems using implicit differentiation.

Question 1:

We have the equation z² + y³ = 10, and we need to find dz/dy without first solving for y.

Differentiating both sides of the equation with respect to y:

2z * dz/dy + 3y² = 0

Rearranging the equation to solve for dz/dy:

dz/dy = -3y² / (2z)

Question 2:

We have the equation z² * y² = 64/81, and we need to find dy/dz.

Differentiating both sides of the equation with respect to z:

2z * y² * dz/dz + z² * 2y * dy/dz = 0

Simplifying the equation and solving for dy/dz:

dy/dz = -2zy / (2y² * z + z²)

Question 3:

We have the inequality 4x² + 3x + 2y <= 110, and we need to find dy/dz.

Since this is an inequality, we cannot directly differentiate it. Instead, we can consider the given point (-5, -5) as a specific case and evaluate the slope at that point.

Substituting x = -5 and y = -5 into the equation, we get:

4(-5)² + 3(-5) + 2(-5) <= 110

100 - 15 - 10 <= 110

75 <= 110

Since the inequality is true, the slope dy/dz exists at the given point.

Question 4:

We have the equation 16 + 1022y + y² = 27, and we need to find dy/dz. Now, we need to find the equation of the tangent line to the curve at (1, 1).

First, differentiate both sides of the equation with respect to z:

0 + 1022 * dy/dz + 2y * dy/dz = 0

Simplifying the equation and solving for dy/dz:

dy/dz = -1022 / (2y)

Question 5:

We have the equation -2x² - 3ry - 2y³ = -76, and we need to find the slope of the tangent line at the point (2, 3).

Differentiating both sides of the equation with respect to x:

-4x - 3r * dy/dx - 6y² * dy/dx = 0

Substituting x = 2, y = 3 into the equation:

-8 - 3r * dy/dx - 54 * dy/dx = 0

Simplifying the equation and solving for dy/dx:

dy/dx = -8 / (3r + 54)

Question 6:

We have the equation 2(x² + y²)² = 25(x² - y²), and we need to find the slope of the tangent line at the point (3, -1).

Differentiating both sides of the equation with respect to x:

4(x² + y²)(2x) = 25(2x - 2y * dy/dx)

Substituting x = 3, y = -1 into the equation:

4(3² + (-1)²)(2 * 3) = 25(2 * 3 - 2(-1) * dy/dx)

Simplifying the equation and solving for dy/dx:

dy/dx = -16 / 61

In some of the questions, we had to substitute specific values to evaluate the slope at a given point because the differentiation alone was not enough to find the slope.

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