Recall from Section 4.2 that the loudness βof a sound in decibels (db) is given by β=10 log(I /

step 1 So to help us find the derivative of this, we need to first review, how do you find the derivati

Recall from Section 4.2 that the loudness βof a sound in...

Recall from Section 4.2 that the loudness $\beta$ of a sound in decibels (db) is given by $\beta=10 \log \left(I / I_{0}\right),$ where $I$ is the intensity of the sound in watts per square meter $\left(\mathrm{W} / \mathrm{m}^{2}\right)$ and $I_{0}$ is a constant that is approximately the intensity of a sound at the threshold of human hearing. Find the rate of change of $\beta$ with respect to $I$ at the point where (a) $I / I_{0}=10$ (b) $I / I_{0}=100$ (c) $I / I_{0}=1000$

Recall from Section 4.2 that the loudness $\beta$ of a sound in decibels (db) is given by $\beta=10 \log \left(I / I_{0}\right),$ where $I$ is the intensity of the sound in watts per square meter $\left(\mathrm{W} / \mathrm{m}^{2}\right)$ and $I_{0}$ is a constant that is approximately the intensity of a sound at the threshold of human hearing. Find the rate of change of $\beta$ with respect to $I$ at the point where
(a) $I / I_{0}=10$
(b) $I / I_{0}=100$
(c) $I / I_{0}=1000$

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Key Concepts

Chain Rule

The Chain Rule is a critical tool in calculus for differentiating composite functions. It states that the derivative of a composite function is the product of the derivative of the outer function (evaluated at the inner function) and the derivative of the inner function. This rule simplifies the process of finding derivatives when a function is nested within another function.

Rate of Change (Derivative)

The rate of change represents how one quantity varies with respect to another, and in calculus, it is quantified using derivatives. Derivatives assess the sensitivity of one variable to changes in another and are essential in understanding the dynamic behavior of systems and functions.

Logarithmic Functions

Logarithmic functions transform multiplicative relationships into additive ones, making them especially useful in measuring quantities that span large ranges. They are fundamental in various fields such as mathematics, physics, and engineering, where processes like sound intensity, earthquake magnitude, and pH levels are described on a logarithmic scale.

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