D - Derivatives Module Overview

Derivatives Module Overview

Introduction

The derivative function is the calculus tool that is used in the workplace to model and predict the behavior of changing quantities such as orbiting spacecraft, growing populations, rising consumer prices, and blood flowing through an artery. Such applications are possible only after a thorough understanding of the relationship between a function and its derivative; rules governing how to calculate derivatives of functions and nonfunctions; techniques for approximating derivatives; interpretations involving slope, tangent lines, and motion; and the connections between differentiability and continuity.

Essential Questions

What is a derivative?
How does a derivative relate to the difference quotient?
How can you use technology's built-in capability to calculate the numerical derivative of a function?
For what types of basic functions do the various differentiation rules apply?
How can a function be transformed prior to differentiation when applying a simpler differentiation rule?
How do you write the equation of a tangent line?
Under what circumstances is a tangent line vertical or undefined?
What is the relationship between differentiability and continuity?
What are the Constant Rule and Constant Multiple Rule and when should they be used?
What is the Product Rule and when should it be used?
What is the Quotient Rule and when should it be used?
What is the Power Rule and when should it be used?
How and when do you use the Chain Rule to take a derivative?
How are the numerical approximation, graphical representation, and analytical determination of a derivative related?
What is meant by higher order derivatives?
What is an explicitly defined function? an implicitly defined function?
How do you use implicit differentiation to find the derivative?
How can rate of change be approximated from graphs and tables of values?
How are instantaneous rate of change at a point and average rate of change related?
In what ways can the derivative be interpreted as an instantaneous rate of change in varied applied contexts?
What role do derivatives play as a foundation for calculus and in practical applications?

Key Terms

The following key terms will help you understand the content in this module.

Acceleration - The rate of change (derivative) of velocity with respect to time.

Average velocity - The ratio of distance traveled (change in distance) to time elapsed (change in time) for a position function $LaTeX: s=f\left(t\right),\:v_{avg}=\frac{\Delta s}{\Delta t}$ $s=f\left(t\right),\:v_{avg}=\frac{\Delta s}{\Delta t}$ . The slope of the secant line.

Chain Rule - A theorem that states how to find the derivative of a composite function. If y is a differentiable function of u and u is a differentiable function of x, then $LaTeX: \frac{dy}{dx}=\frac{dy}{du}=\frac{du}{dx}$ $\frac{dy}{dx}=\frac{dy}{du}=\frac{du}{dx}$

Derivative of a function f with respect to x - The limit of the different of quotient $LaTeX: \frac{f\left(x+\Delta x\right)-f\left(x\right)}{\Delta x}$ $\frac{f\left(x+\Delta x\right)-f\left(x\right)}{\Delta x}$ as $LaTeX: \Delta x$ $\Delta x$ approaches 0. The (instantaneous) rate of change of a function f. On the graph of the curve y=f(x), the slope of the tangent line at the point (x, f(x)).

Difference Quotient - The ratio $LaTeX: \frac{f\left(x+h\right)-f\left(x\right)}{h}$ $\frac{f\left(x+h\right)-f\left(x\right)}{h}$ for the function f at x. It measure the average rate of change of the function f.

Differentiation - The process of determining the derivative of a function.

Explicit function - A relationship that equates the dependent variable directly with a function of the independent variable, as in y = f(x), so that its values may be directly calculated from those of the independent variables.

Higher order derivatives - Derivatives of derivatives.

Implicit differentiation - The computation of the derivative of an implicit function without explicitly determining the function.

Implicit function - A relationship that does not express the value of the dependent variable directly as a function of the independent variable, but rather states a relationship which the variables must jointly satisfy.

Instantaneous velocity - The derivative of the position (distance) function s = f(t). The slope of the tangent line.

Jerk - The rate of change (derivative) of acceleration with respect to time.

Normal line - A line perpendicular to a tangent line to a curve at the point where it intersects the curve.

Numerical derivative (NDeriv) - A numerical method for approximating a derivative at a point that utilizes the symmetric difference quotient. A calculator calculus command that returns the approximate value of a derivative with respect to a given variable at a stated value of X. nDeriv(expression, variable, value)

Power Rule - A specialized case of the Chain Rule for composite functions, $LaTeX: y=\left[u\left(x\right)\right]^n$ $y=\left[u\left(x\right)\right]^n$ . If u is a differentiable function of x and n is any real number, then $LaTeX: \frac{d}{dx}\left(u^n\right)=nu^{n-1}\frac{du}{dx}$ $\frac{d}{dx}\left(u^n\right)=nu^{n-1}\frac{du}{dx}$

Secant line - A line that intersects a curve in two places.

Speed - The absolute value of the velocity of an object.

Tangent line - A line that touches a curve at a point and has the same slope as the curve at the point.

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