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Ch. 3 - Derivatives
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 3 - DerivativesProblem 3.7.1
Chapter 3, Problem 3.7.1

Differentiating Implicitly


Use implicit differentiation to find dy/dx in Exercises 1–14.


x²y + xy² = 6

Verified step by step guidance
1
Start by differentiating both sides of the equation with respect to x. Remember that y is a function of x, so when differentiating terms involving y, use implicit differentiation.
Differentiate the first term x²y. Apply the product rule: d/dx(x²y) = x²(dy/dx) + y(2x).
Differentiate the second term xy². Again, use the product rule: d/dx(xy²) = x(2y(dy/dx)) + y².
Differentiate the right side of the equation, which is a constant: d/dx(6) = 0.
Combine all the differentiated terms: x²(dy/dx) + y(2x) + x(2y(dy/dx)) + y² = 0. Solve for dy/dx by isolating it on one side of the equation.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Implicit Differentiation

Implicit differentiation is a technique used to differentiate equations where the dependent and independent variables are not explicitly separated. Instead of solving for y in terms of x, we differentiate both sides of the equation with respect to x, treating y as a function of x. This allows us to find dy/dx without isolating y, which is particularly useful for complex relationships.
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Finding The Implicit Derivative

Product Rule

The product rule is a fundamental differentiation rule used when differentiating the product of two functions. It states that if u and v are functions of x, then the derivative of their product is given by d(uv)/dx = u'v + uv'. In the context of implicit differentiation, this rule is essential when differentiating terms that involve products of x and y, ensuring that both functions are accounted for.
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The Product Rule

Chain Rule

The chain rule is a key principle in calculus that allows us to differentiate composite functions. It states that if a function y is dependent on u, which in turn is dependent on x, then dy/dx = (dy/du) * (du/dx). In implicit differentiation, the chain rule is applied when differentiating terms involving y, as we must multiply by dy/dx to account for the dependence of y on x.
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Intro to the Chain Rule
Related Practice
Textbook Question

Tolerance The height and radius of a right circular cylinder are equal, so the cylinder’s volume is V = πh³. The volume is to be calculated with an error of no more than 1% of the true value. Find approximately the greatest error that can be tolerated in the measurement of h, expressed as a percentage of h.

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Textbook Question

In Exercises 11–18, find the slope of the function’s graph at the given point. Then find an equation for the line tangent to the graph there.


f(x) = √(x + 1), (8, 3)

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Textbook Question

Differentiating Implicitly


Use implicit differentiation to find dy/dx in Exercises 1–14.


x³ + y³ = 18xy

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Textbook Question

Find the derivatives of the functions in Exercises 19–40.


q = sin(t / (√t + 1))

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Textbook Question

Derivative Calculations


In Exercises 1–8, given y = f(u) and u = g(x), find dy/dx = f'(g(x)) g'(x).


y = sin u, u = 3x + 1

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Textbook Question

Find the points on the curve y = tan x, -π/2 < x < π/2, where the normal line is parallel to the line y = -x/2. Sketch the curve and normal lines together, labeling each with its equation.

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