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Ch. 3 - Derivatives
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 3 - DerivativesProblem 3.8.48b
Chapter 3, Problem 3.8.48b

45–50. Tangent lines Carry out the following steps. <IMAGE>
b. Determine an equation of the line tangent to the curve at the given point.
x⁴-x²y+y⁴=1; (−1, 1)

Verified step by step guidance
1
First, understand that the problem requires finding the equation of the tangent line to the curve defined by the equation x⁴ - x²y + y⁴ = 1 at the point (-1, 1). This involves using implicit differentiation to find the derivative dy/dx.
Differentiate both sides of the equation x⁴ - x²y + y⁴ = 1 with respect to x. Remember to apply the product rule to the term x²y and the chain rule to y⁴. The derivative of x⁴ is 4x³, and the derivative of y⁴ with respect to x is 4y³(dy/dx).
For the term x²y, apply the product rule: the derivative is 2xy + x²(dy/dx). Combine all these derivatives to form the equation: 4x³ - (2xy + x²(dy/dx)) + 4y³(dy/dx) = 0.
Solve the resulting equation for dy/dx, which represents the slope of the tangent line at any point (x, y) on the curve. Substitute the given point (-1, 1) into the equation to find the specific slope at that point.
Once you have the slope, use the point-slope form of the equation of a line, y - y₁ = m(x - x₁), where m is the slope and (x₁, y₁) is the given point (-1, 1), to write the equation of the tangent line.

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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 defined implicitly by an equation involving both x and y. In this case, the equation x⁴ - x²y + y⁴ = 1 requires us to differentiate both sides with respect to x, treating y as a function of x. This allows us to find dy/dx, which is essential for determining the slope of the tangent line at a specific point.
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Finding The Implicit Derivative

Tangent Line Equation

The equation of a tangent line at a given point on a curve can be expressed using the point-slope form: y - y₀ = m(x - x₀), where (x₀, y₀) is the point of tangency and m is the slope of the tangent line. Once the slope is calculated using implicit differentiation, this formula can be applied to find the specific equation of the tangent line at the point (-1, 1) on the curve.
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Equations of Tangent Lines

Slope of the Tangent Line

The slope of the tangent line at a point on a curve represents the instantaneous rate of change of the function at that point. It is calculated as the derivative of the function evaluated at the specific x-coordinate. In this problem, finding the slope at the point (-1, 1) is crucial for constructing the tangent line equation, as it directly influences the line's steepness and direction.
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Slopes of Tangent Lines
Related Practice
Textbook Question

97–100. Logistic growth Scientists often use the logistic growth function P(t) = P₀K / P₀+(K−P₀)e^−r₀t to model population growth, where P₀ is the initial population at time t=0, K is the carrying capacity, and r₀ is the base growth rate. The carrying capacity is a theoretical upper bound on the total population that the surrounding environment can support. The figure shows the sigmoid (S-shaped) curve associated with a typical logistic model. <IMAGE>


{Use of Tech} Gone fishing When a reservoir is created by a new dam, 50 fish are introduced into the reservoir, which has an estimated carrying capacity of 8000 fish. A logistic model of the fish population is P(t) = 400,000 / 50+7950e^−0.5t, where t is measured in years.


b. How long does it take for the population to reach 5000 fish? How long does it take for the population to reach 90% of the carrying capacity?

Textbook Question

Derivatives of inverse functions from a table Use the following tables to determine the indicated derivatives or state that the derivative cannot be determined. <IMAGE>

b. (f^-1)'(6)

Textbook Question

Velocity of a car The graph shows the position s=f(t) of a car t hours after 5:00 P.M. relative to its starting point s=0,where s is measured in miles. <IMAGE>

b. At approximately what time is the car traveling the fastest? The slowest?

Textbook Question

Use a graphing utility to plot the curve and the tangent line.

y = cos x / 1−cos x; x = π/3

Textbook Question

Consider the following cost functions.

b. Determine the average cost and the marginal cost when x=a.

C(x) = − 0.01x²+40x+100, 0≤x≤1500, a=1000

Textbook Question

109-112 {Use of Tech} Calculating limits The following limits are the derivatives of a composite function g at a point a.

b. Use the Chain Rule to find each limit. Verify your answer by using a calculator.

limx→04+sin(x)−2x{\(\displaystyle\]\lim\)_{x\(\to\)0}}\(\frac{\sqrt{4+\sin\left(x\right)}\)-2}{x}