Textbook Question
Find the derivatives of the functions in Exercises 19–40.
q = sin(t / (√t + 1))
Ch. 3 - Derivatives
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 3, Problem 3.7.9
Differentiating Implicitly
Use implicit differentiation to find dy/dx in Exercises 1–14.
x = sec y
Verified step by step guidance1
Start by understanding implicit differentiation. This technique is used when you have an equation involving both x and y, and you want to find dy/dx without solving for y explicitly.
Given the equation x = sec(y), differentiate both sides with respect to x. Remember that when differentiating y with respect to x, you need to multiply by dy/dx due to the chain rule.
Differentiate the left side: The derivative of x with respect to x is 1.
Differentiate the right side: The derivative of sec(y) with respect to y is sec(y)tan(y). Since y is a function of x, apply the chain rule and multiply by dy/dx.
Set the derivatives equal to each other: 1 = sec(y)tan(y) * dy/dx. 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 one variable in terms of the other, we differentiate both sides of the equation with respect to the independent variable, applying the chain rule as necessary. This method is particularly useful for equations that define y implicitly in terms of x.
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Finding The Implicit Derivative
Chain Rule
The chain rule is a fundamental principle in calculus that allows us to differentiate composite functions. It states that if a function y is defined as a function of u, which in turn is a function of x, then the derivative of y with respect to x can be found by multiplying the derivative of y with respect to u by the derivative of u with respect to x. This rule is essential when differentiating implicitly, as it helps manage the relationships between variables.
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05:02Intro to the Chain Rule
Secant Function
The secant function, denoted as sec(y), is the reciprocal of the cosine function, defined as sec(y) = 1/cos(y). In the context of the given equation x = sec(y), it relates the angle y to the horizontal distance x in a right triangle. Understanding the properties of the secant function is crucial for implicit differentiation, as it influences how we differentiate the equation with respect to x.
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Related Practice
Textbook Question
Second Derivatives
Find y'' in Exercises 59–64.
y = (1 − √x)⁻¹
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 = x − cos x
Textbook Question
Find the value of a that makes the following function differentiable for all x-values.
g(x) = { ax, if x < 0
x² − 3x, if x ≥ 0
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.
h(t) = t³ + 3t, (1, 4)
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