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Derivatives using tables Let and . Use the table to compute the following derivatives.
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b.
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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 3.8.20b
13-26 Implicit differentiation Carry out the following steps.
b. Find the slope of the curve at the given point.
tan xy = x+y; (0,0)
Guida verificata passo dopo passo1
Start by differentiating both sides of the equation with respect to x. The equation is \( \tan(xy) = x + y \). Use implicit differentiation, which involves differentiating each term while treating y as a function of x.
Differentiate the left side: The derivative of \( \tan(u) \) with respect to u is \( \sec^2(u) \), so apply the chain rule: \( \frac{d}{dx}[\tan(xy)] = \sec^2(xy) \cdot \frac{d}{dx}(xy) \).
Apply the product rule to differentiate \( xy \): \( \frac{d}{dx}(xy) = x \frac{dy}{dx} + y \). Therefore, the derivative of the left side becomes \( \sec^2(xy) \cdot (x \frac{dy}{dx} + y) \).
Differentiate the right side: The derivative of \( x + y \) with respect to x is \( 1 + \frac{dy}{dx} \).
Set the derivatives equal to each other: \( \sec^2(xy) \cdot (x \frac{dy}{dx} + y) = 1 + \frac{dy}{dx} \). Solve this equation for \( \frac{dy}{dx} \) to find the slope of the curve at the point (0,0). Substitute x = 0 and y = 0 into the equation to find the specific slope at that point.
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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 like 'tan(xy) = x + y', where y cannot be easily isolated.
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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 function at that point, which is mathematically expressed as the derivative. For a curve defined implicitly, the slope can be found by evaluating the derivative obtained through implicit differentiation. In this case, we will find the slope at the point (0,0) by substituting these coordinates into the derivative expression.
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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 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 in implicit differentiation, where we often need to differentiate terms involving products of variables.
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05:02Intro to the Chain Rule
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{Use of Tech} Angle of elevation A small plane, moving at 70 m/s, flies horizontally on a line 400 meters directly above an observer. Let θ be the angle of elevation of the plane (see figure). <IMAGE>
b. Graph dθ/dx as a function of x and determine the point at which θ changes most rapidly.
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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
The energy (in joules) released by an earthquake of magnitude M is given by the equation E=25,000 ⋅ 101.5M. (This equation can be solved for M to define the magnitude of a given earthquake; it is a refinement of the original Richter scale created by Charles Richter in 1935.)
Compute dE/dM and evaluate it for M=3. What does this derivative mean? (M has no units, so the units of the derivative are J per change in magnitude.)
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Domanda del libro di testo
A differential equation is an equation involving an unknown function and its derivatives. Consider the differential equation y′′(t)+y(t) = 0.
b. Show that y = B cos t satisfies the equation for any constant B.
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