Domanda del libro di testo
Find an equation of the line tangent to the following curves at the given value of x.
y = 4 sin x cos x; x = π/3
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Ch. 3 - Derivatives
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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TOC.tableOfContents
- 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: 9780136847243Non è quello che usi tu?Cambia libro di testo
Capitolo 3, Problema 3.8.17a
13-26 Implicit differentiation Carry out the following steps.
a. Use implicit differentiation to find dy/dx.
sin y = 5x⁴−5; (1, π)
Guida verificata passo dopo passo1
Start by differentiating both sides of the equation with respect to x. The equation is \( \sin y = 5x^4 - 5 \).
For the left side, differentiate \( \sin y \) with respect to y, which gives \( \cos y \), and then multiply by \( \frac{dy}{dx} \) due to the chain rule. This results in \( \cos y \cdot \frac{dy}{dx} \).
For the right side, differentiate \( 5x^4 - 5 \) with respect to x. The derivative of \( 5x^4 \) is \( 20x^3 \), and the derivative of \(-5\) is 0. So, the right side becomes \( 20x^3 \).
Set the derivatives equal to each other: \( \cos y \cdot \frac{dy}{dx} = 20x^3 \).
Solve for \( \frac{dy}{dx} \) by dividing both sides by \( \cos y \), resulting in \( \frac{dy}{dx} = \frac{20x^3}{\cos y} \).
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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.
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Percorso guidato
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 terms involving y, resulting in the derivative dy/dx. This means that when differentiating a function of y, we multiply by dy/dx to account for the dependence of y on x.
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05:02Intro to the Chain Rule
Evaluating at a Point
After finding the derivative dy/dx using implicit differentiation, we often need to evaluate it at a specific point, such as (1, π) in this case. This involves substituting the x and y values into the derived expression to find the slope of the tangent line at that point. This step is crucial for understanding the behavior of the function at specific coordinates.
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04:50Critical Points
Pratica correlata
Domanda del libro di testo
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)
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Domanda del libro di testo
7–14. Find the derivative the following ways:
a. Using the Product Rule (Exercises 7–10) or the Quotient Rule (Exercises 11–14). Simplify your result.
f(w) = w³ -w / w
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Domanda del libro di testo
Airline travel The following figure shows the position function of an airliner on an out-and-back trip from Seattle to Minneapolis, where s = f(t) is the number of ground miles from Seattle t hours after take-off at 6:00 A.M. The plane returns to Seattle 8.5 hours later at 2:30 P.M. <IMAGE>
a. Calculate the average velocity of the airliner during the first 1.5 hours of the trip (0 ≤ t ≤ 1.5).
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Domanda del libro di testo
The volume V of a sphere of radius r changes over time t.
a. Find an equation relating dV/dt to dr/dt.
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Domanda del libro di testo
{Use of Tech} Approximating derivatives Assuming the limit exists, the definition of the derivative f′(a) = lim h→0 f(a + h) − f(a) / h implies that if ℎ is small, then an approximation to f′(a) is given by
f' (a) ≈ f(a+h) - f(a) / h. If ℎ > 0 , then this approximation is called a forward difference quotient; if ℎ < 0 , it is a backward difference quotient. As shown in the following exercises, these formulas are used to approximate f′ at a point when f is a complicated function or when f is represented by a set of data points. <IMAGE>
Let f (x) = √x.
a. Find the exact value of f' (4).
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