Textbook Question
45–50. Tangent lines Carry out the following steps. <IMAGE>
a. Verify that the given point lies on the curve.
x³+y³=2xy; (1, 1)
Ch. 3 - Derivatives
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 3, Problem 3.8.19a
13-26 Implicit differentiation Carry out the following steps.
a. Use implicit differentiation to find dy/dx.
cos y = x; (0, π/2)
Verified step by step guidance1
Start by differentiating both sides of the equation with respect to x. The equation is cos(y) = x.
Differentiate the left side: The derivative of cos(y) with respect to x is -sin(y) * (dy/dx) because of the chain rule.
Differentiate the right side: The derivative of x with respect to x is 1.
Set the derivatives equal to each other: -sin(y) * (dy/dx) = 1.
Solve for dy/dx by isolating it: dy/dx = -1/sin(y).
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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 method is particularly useful for equations that are difficult or impossible to rearrange.
Recommended video:
05:14
Finding The Implicit Derivative
Chain Rule
The chain rule is a fundamental principle in calculus that allows us to differentiate composite functions. When using implicit differentiation, the chain rule is applied to account for the derivative of y with respect to x, denoted as dy/dx. This means that when differentiating a function of y, we multiply by dy/dx to reflect the dependence of y on x.
Recommended video:
05:02Intro to the Chain Rule
Trigonometric Functions
Trigonometric functions, such as sine, cosine, and tangent, are essential in calculus for modeling periodic phenomena and solving various problems. In the context of the given equation, cos(y) = x, understanding the properties and derivatives of these functions is crucial for finding dy/dx. The derivative of cos(y) involves the chain rule and is equal to -sin(y) * dy/dx.
Recommended video:
6:04Introduction to Trigonometric Functions
Related Practice
Textbook Question
Surface area of a cone The lateral surface area of a cone of radius r and height h (the surface area excluding the base) is A = πr√r²+h².
a. Find dr/dh for a cone with a lateral surface area of A=1500π.
Textbook Question
Derivatives and tangent lines
a. For the following functions and values of a, find f′(a).
f(x) = √3x; a= 12
Textbook Question
Deriving trigonometric identities
a. Differentiate both sides of the identity cos 2t = cos² t−sin² t to prove that sin 2 t= 2 sin t cos t.
Textbook Question
{Use of Tech} Tree growth Let b represent the base diameter of a conifer tree and let h represent the height of the tree, where b is measured in centimeters and h is measured in meters. Assume the height is related to the base diameter by the function h = 5.67+0.70b+0.0067b².
a. Graph the height function.
Textbook Question
21–30. Derivatives
a. Use limits to find the derivative function f' for the following functions f.
f(t) = 3t⁴; a= -2, 2
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