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Ch. 7 - Transcendental Functions
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 7 - Transcendental FunctionsProblem 7.3.61
Chapter 7, Problem 7.3.61

In Exercises 59–86, find the derivative of y with respect to the given independent variable.
61. y = 5√s

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Identify the given function: \( y = 5\sqrt{s} \). Recall that \( \sqrt{s} \) can be rewritten as \( s^{\frac{1}{2}} \).
Rewrite the function using exponent notation: \( y = 5s^{\frac{1}{2}} \). This makes it easier to apply the power rule for differentiation.
Apply the constant multiple rule: The derivative of \( 5s^{\frac{1}{2}} \) is \( 5 \) times the derivative of \( s^{\frac{1}{2}} \).
Use the power rule for derivatives: For \( s^{n} \), the derivative with respect to \( s \) is \( n s^{n-1} \). So, \( \frac{d}{ds} s^{\frac{1}{2}} = \frac{1}{2} s^{-\frac{1}{2}} \).
Combine the results: Multiply the constant 5 by the derivative from the power rule to get \( \frac{dy}{ds} = 5 \times \frac{1}{2} s^{-\frac{1}{2}} \).

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Key Concepts

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

Derivative of a Power Function

The derivative of a function in the form y = s^n is found using the power rule, which states that dy/ds = n * s^(n-1). This rule simplifies differentiation of roots and powers by expressing roots as fractional exponents.
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Representing Functions as Power Series

Rewriting Roots as Exponents

A root such as the square root of s can be rewritten as s^(1/2). This conversion allows the use of the power rule for differentiation, making it easier to handle roots in calculus problems.
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Guided course
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Introduction to Exponent Rules

Constant Multiple Rule

When differentiating a function multiplied by a constant, the constant can be factored out and multiplied by the derivative of the variable part. For example, if y = 5 * s^(1/2), then dy/ds = 5 * d/ds(s^(1/2)).
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Related Practice
Textbook Question

25. First-order chemical reactions In some chemical reactions, the rate at which the amount of a substance changes with time is proportional to the amount present. For the change of δ-gluconolactone into gluconic acid, for example,

dy/dt = -0.6y

when t is measured in hours. If there are 100 grams of δ-gluconolactone present when t=0, how many grams will be left after the first hour?

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