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Applied Calculus

Differentiation: The Language of Change

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1
The Derivative of a Function and Two Interpretations

**Introduction and Definitions**

What is the definition of the derivative?

If a function represents distance as a function of time, what does its derivative represent?

If a function represents the cost to produce \(x\) items, what does its derivative cost?

If the volume of a sphere is a function of its radius, what is the relationship between the rate of change of the volume and the rate of change of the radius?

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2
Differentiating Products and Quotients

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3
Higher Order Derivatives

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4
The Chain Rule and General Power Rule

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5
Implicit Differentiation

Suppose both \(y\) and \(x\) are differentiable functions of \(t\) and that the relationship between \(y\) and \(x\) is expressed by the equation \(4x^3+3y^5=960\). Find and interpret \(\displaystyle\frac{dy}{dt}\) when \(\displaystyle\frac{dx}{dt}=4\), \(x=6\), and \(y=2\).

Use the function \(x=\displaystyle\frac{20000}{\sqrt[3]{2p^2-5}}+350\) to find the rate at which the number of instruments sold is changing with respect to time, when the price of an instrument is \(\$400\) and is changing at a rate of \(\$1\) per month.