Skip to main content
Ch. 3 - Derivatives
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBriggs 3rd EditionCh. 3 - DerivativesProblema 3.8.17b
Capitolo 3, Problema 3.8.17b

13-26 Implicit differentiation Carry out the following steps.
b. Find the slope of the curve at the given point.
sin y = 5x⁴−5; (1, π)

Guida verificata passo dopo passo
1
Step 1: Start with the given equation \( \sin y = 5x^4 - 5 \). This equation involves both \( x \) and \( y \), so we'll use implicit differentiation to find \( \frac{dy}{dx} \).
Step 2: Differentiate both sides of the equation with respect to \( x \). Remember that \( y \) is a function of \( x \), so when differentiating \( \sin y \), use the chain rule: \( \frac{d}{dx}(\sin y) = \cos y \cdot \frac{dy}{dx} \).
Step 3: Differentiate the right side of the equation \( 5x^4 - 5 \) with respect to \( x \). The derivative of \( 5x^4 \) is \( 20x^3 \), and the derivative of a constant \(-5\) is 0.
Step 4: Set the derivatives equal to each other: \( \cos y \cdot \frac{dy}{dx} = 20x^3 \). Solve for \( \frac{dy}{dx} \) to find the expression for the slope of the curve.
Step 5: Substitute the given point \( (1, \pi) \) into the expression for \( \frac{dy}{dx} \). Use \( x = 1 \) and \( y = \pi \) to find the slope at this specific point.

Risposta video verificata per un problema simile:

Questa soluzione video è stata consigliata dai nostri tutor come utile per risolvere questo problema.
Durata del video:
2m

Concetti chiave

Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.

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 curves defined by equations that cannot be easily rearranged.
Video consigliato:
Percorso guidato
05:14
Finding The Implicit Derivative

Slope of a Curve

The slope of a curve at a given point represents the rate of change of the dependent variable with respect to the independent variable at that point. Mathematically, it is found by evaluating the derivative of the function at the specified coordinates. In the context of implicit differentiation, the slope can be determined by substituting the coordinates of the point into the derivative obtained from the implicit differentiation process.
Video consigliato:
11:41
Summary of Curve Sketching

Chain Rule

The chain rule is a fundamental principle in calculus used to differentiate composite functions. It states that if a function is composed of two or more functions, the derivative of the outer function is multiplied by the derivative of the inner function. In implicit differentiation, the chain rule is applied when differentiating terms involving the dependent variable, ensuring that the derivative accounts for the relationship between the variables.
Video consigliato:
05:02
Intro to the Chain Rule
Pratica correlata
Domanda del libro di testo

Owlet talons Let L (t) equal the average length (in mm) of the middle talon on an Indian spotted owlet that is t weeks old, as shown in the figure.<IMAGE>

b. Estimate the value of L'(a) for a ≥ 4 . What does this tell you about the talon lengths on these birds? (Source: ZooKeys, 132, 2011)

178
views
Domanda del libro di testo

An object oscillates along a vertical line, and its position in centimeters is given by y(t) = 30(sint - 1), where t ≥ 0 is measured in seconds and y is positive in the upward direction.

Find the velocity of the oscillator, v(t) =y′(t).

274
views
Domanda del libro di testo

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?

253
views
Domanda del libro di testo

13-26 Implicit differentiation Carry out the following steps.

b. Find the slope of the curve at the given point.

cos y = x; (0, π/2)

192
views
Domanda del libro di testo

{Use of Tech} Power and energy The total energy in megawatt-hr (MWh) used by a town is given by E(t) = 400t+2400/π sin πt/12, where t≥0 is measured in hours, with t=0 corresponding to noon.

b. At what time of day is the rate of energy consumption a maximum? What is the power at that time of day?

283
views
Domanda del libro di testo

{Use of Tech} Fuel economy Suppose you own a fuel-efficient hybrid automobile with a monitor on the dashboard that displays the mileage and gas consumption. The number of miles you can drive with g gallons of gas remaining in the tank on a particular stretch of highway is given by m(g) = 50g−25.8g²+12.5g³−1.6g⁴, for 0≤g≤4.

b. Graph and interpret the gas mileage m(g)/g. 

241
views