Domanda del libro di testo
60–62. {Use of Tech} Multiple tangent lines Complete the following steps. <IMAGE>
b. Graph the tangent lines on the given graph.
x+y³−y=1; x=1
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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 3.8.26b
13-26 Implicit differentiation Carry out the following steps.
b. Find the slope of the curve at the given point.
(x+y)^2/3=y; (4, 4)
Guida verificata passo dopo passo1
Start by differentiating both sides of the equation \((x + y)^{2/3} = y\) with respect to \(x\). Remember that \(y\) is a function of \(x\), so you'll need to use implicit differentiation.
Apply the chain rule to differentiate the left side: \(\frac{d}{dx}((x + y)^{2/3}) = \frac{2}{3}(x + y)^{-1/3} \cdot (1 + \frac{dy}{dx})\).
Differentiate the right side with respect to \(x\): \(\frac{d}{dx}(y) = \frac{dy}{dx}\).
Set the derivatives equal to each other: \(\frac{2}{3}(x + y)^{-1/3} \cdot (1 + \frac{dy}{dx}) = \frac{dy}{dx}\).
Solve for \(\frac{dy}{dx}\) by isolating it on one side of the equation. Substitute \(x = 4\) and \(y = 4\) into the equation to find the slope of the curve at the point \((4, 4)\).
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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 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.
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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.
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11:41Summary 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.
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05:02Intro to the Chain Rule
Pratica correlata
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.
³√x+³√y⁴ = 2;(1,1)
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Domanda del libro di testo
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
b. d/dx(tan^−1 x) =sec² x
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Domanda del libro di testo
Derivatives and tangent lines
b. Determine an equation of the line tangent to the graph of f at the point (a,f(a)) for the given value of a.
f(x) = √3x; a= 12
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Domanda del libro di testo
{Use of Tech} A mixing tank A 500-liter (L) tank is filled with pure water. At time t=0, a salt solution begins flowing into the tank at a rate of 5 L/min. At the same time, the (fully mixed) solution flows out of the tank at a rate of 5.5 L/min. The mass of salt in grams in the tank at any time t≥0 is given by M(t) = 250(1000−t)(1−10−³⁰(1000−t)¹⁰) and the volume of solution in the tank is given by V(t) = 500-0.5t.
b. Graph the volume function and verify that the tank is empty when t=1000 min.
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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.
x = e^y; (2, ln 2)
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