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Chapter 3: Problem 22

Find \(d y / d x\) evaluated at the given values. \(x^{2}+y^{2}=25\) at \(x=-3, y=4\)

Short Answer

Expert verified
\(\frac{dy}{dx}\) at \(x=-3, y=4\) is \(\frac{3}{4}\).

Step by step solution

01

Identify Known Equation

We have the equation of a circle: \(x^2 + y^2 = 25\). The goal is to find \(\frac{dy}{dx}\) at the point \((x, y) = (-3, 4)\).
02

Implicit Differentiation

Differentiate both sides of the equation \(x^2 + y^2 = 25\) with respect to \(x\). This gives us \(2x + 2y \frac{dy}{dx} = 0\).
03

Solve for \(\frac{dy}{dx}\)

Rearrange the equation from Step 2 to solve for \(\frac{dy}{dx}\): \[ 2y \frac{dy}{dx} = -2x \]Divide both sides by \(2y\): \[ \frac{dy}{dx} = -\frac{x}{y} \]
04

Evaluate the Derivative at the Given Point

Substitute \(x = -3\) and \(y = 4\) into the expression from Step 3:\[ \frac{dy}{dx} = -\frac{-3}{4} = \frac{3}{4} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Derivative
The derivative represents the rate at which a function changes as its input changes. In simpler terms, it measures how quickly or slowly the value of a function is moving at any given point. In calculus, the derivative of a function is denoted as \(\frac{dy}{dx}\) or \(f'(x)\), which signifies the rate of change of \(y\) with respect to \(x\).
When applied to curves or points in 2D space, the derivative can tell us the slope or angle of the tangent line at any particular point on a curve. This is highly useful in understanding how a function behaves visually and dynamically.
Equation of a Circle
The equation of a circle in a 2D plane is typically represented as \(x^2 + y^2 = r^2\), where \(r\) is the radius of the circle. In our exercise, the circle's equation is \(x^2 + y^2 = 25\), indicating a circle centered at the origin (0,0) with a radius of 5 (since \(r^2 = 25\) means \(r = \sqrt{25} = 5\)).
鉁� Why is this useful?
  • The structure of this equation tells us every point \((x, y)\) that lies on the circle.
  • We can easily visualize or draw a circle using its equation once the radius and center are known.
Points on the circle, like \((x, y) = (-3, 4)\) as given, abide by this equation. Understanding the equation assists in performing other calculus operations like finding derivatives, especially when the shape or form of a graph is involved.
Differentiation Rules
Differentiation is the process of finding the derivative of a function. There are several rules and techniques to perform differentiation efficiently, such as:
  • Power Rule: If \(f(x) = x^n\), then its derivative \(f'(x) = nx^{n-1}\).
  • Product Rule: Used when differentiating products of two functions. If \(u\) and \(v\) are functions of \(x\), then \((uv)' = u'v + uv'\).
  • Quotient Rule: For differentiating ratios of two functions. If \(u\) and \(v\) are functions of \(x\), the rule is \(\left( \frac{u}{v} \right)' = \frac{u'v - uv'}{v^2}\).
  • Chain Rule: For differentiating composite functions. If \(y = f(g(x))\), then \(\frac{dy}{dx} = f'(g(x)) \cdot g'(x)\).
In the context of implicit differentiation, used for our equation of a circle, each term of an equation is treated as a function of \(x\), and these rules are applied to differentiate both sides of the equation accordingly. This allows us to solve for \(\frac{dy}{dx}\), even when \(y\) is not isolated in the equation.

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Most popular questions from this chapter

The number of traffic accidents per year in a city of population \(p\) is predicted to be \(T=0.002 p^{3 / 2}\). If the population is growing by 500 people a year, find the rate at which traffic accidents will be rising when the population is \(p=40,000\).

\(X\) and \(y\) are functions of \(t\) Differentiate with respect to \(t\) to find a relation between \(d x / d t\) and \(d y / d t\). \(x^{3}+y^{2}=1\)

A management consultant estimates that the number \(h\) of hours per day that employees will work and their daily pay of \(p\) dollars are related by the equation \(60 h^{5}+\) \(2,000,000=p^{3}\). Find \(d h / d p\) at \(p=200\) and interpret your answer.

Suppose that you have a formula that relates the value of an investment (denoted by \(x\) ) to the day of the year (denoted by \(y\) ). State, in everyday language, what \(\frac{d y}{d x}\) and \(\frac{d x}{d y}\) would mean.

Find \(d y / d x\) evaluated at the given values. \(x^{2}+y^{2}=x y+7\) at \(x=3, y=2\)
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