Intro to derivatives
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A derivative is defined as the limit of the average slope of a function over an interval as the interval approaches zero. The derivative of a function f(x) at a point a is the limit as h approaches 0 of the difference quotient (f(a+h)-f(a))/h. Taking the derivative of 4x^3 at x=3 using this definition yields 108, which matches taking the derivative of 4x^3 as a function of x and plugging in 3, giving 12x^2 and then 108 at x=3. While the definition can derive many derivatives, some functions require other techniques beyond the limit definition.
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This document provides an introduction to the concept of the tangent line and derivative. It defines the tangent line as the line that intersects a curve at exactly one point. It discusses how to approximate the slope of a tangent line using secant lines and taking the limit as the second point approaches the point of tangency. The derivative is defined as the formula that gives the slope of the tangent line at any point on a curve. It provides examples of using calculators to calculate derivatives and discusses how the graph of a derivative relates to properties of the original function such as maxima, minima and x-intercepts.
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Intro to derivatives
- 1. A derivative is a special kind of limit. The derivative at x a of some function is the limit as h approaches 0 of the average slope of the function over the interval a,(a h) . A derivative function is simply these limits as a function of some reference variable (such as x if the derivative is taken with respect to x). For example, the derivative of the function 4 x 3 at x 3 is: h 0 lim 4(3 h) 3 4(3) 3 (3 h ) 3 h 0 lim 4(3 h) 3 108 h h 0 lim 4(h 3 15 h 2 27 h 27 ) 108 h h 0 lim 4 h 3 60 h 2 108 h h h 0 lim 4 h 2 60 h 108 108 , but using the same process to find the derivative as a function of x of f(x) = 4x3 by not plugging in 3 for x, we get 12x2, and then can plug in 3 for x and get the same answer, 108. This result was achieved by simply using the definition of the derivative at a point and then cancelling the h in the denominator so we can directly take the limit by plugging in 0 for the remaining h’s.
- 2. The derivatives of many functions can be derived using the basic definition of a derivative, shown in the picture below. In the picture to the right, df/dx represents the derivative of f(x) with respect to x. Some types of functions cannot be differentiated through sole use of the definition of the derivative. Sometimes other means are required. Differentiation is the process of taking derivatives.