Textbook Question
For problems 49–52 use implicit differentiation to find dy/dx at the given point P.
51. y arccos(xy) = -3√2/4 π; P(1/2, -√2)
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Ch. 7 - Transcendental Functions
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 7, Problem 7.3.3
In Exercises 1–4, solve for t.
e^(sqrt(t)) = x^2
Verified step by step guidance1
Start with the given equation: \(e^{\sqrt{t}} = x^2\).
Take the natural logarithm (ln) of both sides to eliminate the exponential: \(\ln\left(e^{\sqrt{t}}\right) = \ln\left(x^2\right)\).
Use the logarithm property \(\ln\left(e^a\right) = a\) to simplify the left side: \(\sqrt{t} = \ln\left(x^2\right)\).
Apply the logarithm power rule on the right side: \(\sqrt{t} = 2 \ln(x)\).
Square both sides to solve for \(t\): \(t = \left(2 \ln(x)\right)^2\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Exponential Functions
An exponential function involves a constant base raised to a variable exponent, such as e^(sqrt(t)). Understanding how to manipulate and invert these functions is essential for solving equations where the variable is in the exponent.
Recommended video:
6:13
Exponential Functions
Square Root and Its Properties
The square root function, denoted sqrt(t), is the inverse of squaring a number. Recognizing how to isolate and handle the square root in an equation is crucial for simplifying and solving for the variable t.
Recommended video:
06:21Properties of Functions
Logarithmic Functions and Their Inverses
Logarithms are the inverse operations of exponentials. Applying the natural logarithm (ln) to both sides of an equation like e^(sqrt(t)) = x^2 allows us to 'undo' the exponential and solve for sqrt(t), facilitating the isolation of t.
Recommended video:
5:26Graphs of Logarithmic Functions
Related Practice
Textbook Question
L’Hôpital’s Rule
Find the limits in Exercises 103–110.
105. lim(x→∞) x arctan(2/x)
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Textbook Question
In Exercises 27–32, find dy/dx.
e^(2x)=sin(x+3y)
Textbook Question
Use l’Hôpital’s rule to find the limits in Exercises 7–52.
17. lim (θ → π/2) (2θ - π) / cos(2π - θ)
Textbook Question
Evaluate the integrals in Exercises 111–114.
113. ∫₁^(1/x) (1 / t) dt,x > 0
Textbook Question
Solve the differential equation in Exercises 9–22.
19. y²(dy/dx) = 3x²y³ - 6x²
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