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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 3.8.5
Chapter 3, Problem 3.8.5

If y = x² and dx/dt = 3, then what is dy/dt when x = –1?

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1
First, identify the given function: y = x². This is the function that relates y and x.
Recognize that you need to find dy/dt, which is the rate of change of y with respect to time t.
Use the chain rule for differentiation, which states that dy/dt = (dy/dx) * (dx/dt).
Differentiate y = x² with respect to x to find dy/dx. The derivative of x² with respect to x is 2x.
Substitute the given values into the chain rule formula: dy/dt = 2x * (dx/dt). Use x = -1 and dx/dt = 3 to find dy/dt.

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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 find the derivative of a function when it is not explicitly solved for one variable in terms of another. In this problem, it helps us differentiate y = x² with respect to time t, considering x as a function of t, to find dy/dt.
Recommended video:
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Finding The Implicit Derivative

Chain Rule

The chain rule is a fundamental principle in calculus used to differentiate composite functions. It states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function times the derivative of the inner function. Here, it allows us to relate dy/dt to dx/dt by differentiating y = x² with respect to t.
Recommended video:
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Intro to the Chain Rule

Related Rates

Related rates problems involve finding the rate at which one quantity changes with respect to time, given the rate of change of another related quantity. In this scenario, we are given dx/dt and need to find dy/dt when x = -1, using the relationship between x and y provided by the equation y = x².
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Intro To Related Rates
Related Practice
Textbook Question

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


g(t) = (1 + sin(3t) / (3 − 2t))⁻¹

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Textbook Question

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


y = (5 − 2x)⁻³ + (1 / 8)(2 / x + 1)⁴

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In Exercises 1–12, find the first and second derivatives.


w = 3z⁷ − 7z³ + 21z²

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In Exercises 29 and 30, find the slope of the curve at the given points.


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Textbook Question

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In Exercises 29–34, each function f(x) changes value when x changes from x₀ to x₀ + dx. Find


a. the change Δf = f(x₀ + dx) − f(x₀);

b. the value of the estimate df = fʹ(x₀) dx; and

c. the approximation error |Δf − df|.


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f(x) = x² + 2x, x₀ = 1, dx = 0.1

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Textbook Question

One-Sided Derivatives


Compute the right-hand and left-hand derivatives as limits to show that the functions in Exercises 37–40 are not differentiable at the point P.

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