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Ch. 3 - Derivatives
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 3 - DerivativesProblem 3.6.19
Chapter 3, Problem 3.6.19

Find the derivatives of the functions in Exercises 19–40.


p = √(3 − t)

Verified step by step guidance
1
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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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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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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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Intro 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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Completing the Square to Rewrite the Integrand Example 6
Related Practice
Textbook Question

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?

Textbook Question

In Exercises 41–58, find dy/dt.


y = √(3t + (√2 + √(1 − t)))

Textbook Question

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)

Textbook Question

Differentiating Implicitly


Use implicit differentiation to find dy/dx in Exercises 1–14.


x⁴ + sin y = x³y²

Textbook Question

Assume that functions f and g are differentiable with f(2) = 3, f'(2) = −1, g(2) = −4, and g'(2) = 1. Find an equation of the line perpendicular to the line tangent to the graph of F(x) = (f(x) + 3) / (x − g(x)) at x = 2.

Textbook Question

Suppose that functions ƒ(x) and g(x) and their first derivatives have the following values at x = 0 and x = 1.


x ƒ(x) g(x) ƒ'(x) g'(x)

0 1 1 -3 1/2

1 3 5 1/2 -4


Find the first derivatives of the following combinations at the given value of x.


a. 6ƒ(x) - g(x), x = 1

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