Domanda del libro di testo
Theory and Examples
Intersecting normal line The line that is normal to the curve x² + 2xy – 3y² = 0 at (1,1) intersects the curve at what other point?
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Ch. 3 - Derivatives
Hass15th EditionThomas' CalculusISBN: 9780137616077
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TOC.tableOfContents
- 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 Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
Capitolo 3, Problema 3.6.19
Find the derivatives of the functions in Exercises 19–40.
p = √(3 − t)
Guida verificata passo dopo passo1
Identify the function you need to differentiate: \( p = \sqrt{3 - t} \). This can be rewritten using exponent notation as \( p = (3 - t)^{1/2} \).
Apply the chain rule for differentiation. The chain rule states that if you have a composite function \( f(g(t)) \), then the derivative \( f'(g(t)) \cdot g'(t) \) is used.
Differentiate the outer function \( f(u) = u^{1/2} \) with respect to \( u \). The derivative is \( \frac{1}{2}u^{-1/2} \).
Differentiate the inner function \( g(t) = 3 - t \) with respect to \( t \). The derivative is \( -1 \).
Combine the derivatives using the chain rule: \( \frac{d}{dt} p = \frac{1}{2}(3 - t)^{-1/2} \cdot (-1) \). Simplify the expression to find the derivative of \( p \).
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Derivative
The derivative of a function measures how the function's output value changes as its input value changes. It is defined as the limit of the average rate of change of the function over an interval as the interval approaches zero. In practical terms, the derivative provides the slope of the tangent line to the function's graph at any given point.
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Percorso guidato
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Derivatives
Chain Rule
The chain rule is a fundamental technique in calculus used to differentiate composite functions. It states that if a function y is composed of two functions u and t, then the derivative of y with respect to t can be found by multiplying the derivative of y with respect to u by the derivative of u with respect to t. This rule is essential for functions that are nested within one another.
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05:02Intro to the Chain Rule
Square Root Function
The square root function, denoted as √x, is a function that returns the non-negative value whose square is x. When differentiating functions involving square roots, it is important to apply the chain rule correctly, as the square root can be expressed as a power of one-half. Understanding the properties of square roots is crucial for accurately finding derivatives of such functions.
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03:19Completing the Square to Rewrite the Integrand Example 6
Pratica correlata
Domanda del libro di testo
In Exercises 41–58, find dy/dt.
y = √(3t + (√2 + √(1 − t)))
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Domanda del libro di testo
Finding Derivative Functions and Values
Using the definition, calculate the derivatives of the functions in Exercises 1–6. Then find the values of the derivatives as specified.
g(t) = 1/t²; g′(−1), g′(2), g′(√3)
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Domanda del libro di testo
Differentiating Implicitly
Use implicit differentiation to find dy/dx in Exercises 1–14.
x⁴ + sin y = x³y²
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Domanda del libro di testo
Find the derivatives of the functions in Exercises 17–28.
y = ((x + 1)(x + 2)) / ((x − 1)(x − 2))
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Domanda del libro di testo
A growing sand pile Sand falls from a conveyor belt at the rate of 10 m³/min onto the top of a conical pile. The height of the pile is always three-eighths of the base diameter. How fast are the (a) height and (b) radius changing when the pile is 4 m high? Answer in centimeters per minute.
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