{"id":2816,"date":"2024-06-10T16:35:23","date_gmt":"2024-06-10T16:35:23","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus1\/?post_type=chapter&p=2816"},"modified":"2024-08-05T12:33:25","modified_gmt":"2024-08-05T12:33:25","slug":"defining-the-derivative-learn-it-2","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus1\/chapter\/defining-the-derivative-learn-it-2\/","title":{"raw":"Defining the Derivative: Learn It 2","rendered":"Defining the Derivative: Learn It 2"},"content":{"raw":"

The Derivative of a Function at a Point<\/h2>\r\n

The type of limit we compute to find the slope of the tangent line to a function at a point has many applications across various disciplines. These include velocity and acceleration in physics, marginal profit functions in business, and growth rates in biology. This limit occurs so frequently that we give it a special name: the derivative<\/strong>. The process of finding a derivative is called differentiation<\/strong>.<\/p>\r\n

\r\n

derivative<\/h3>\r\n

Let [latex]f(x)[\/latex] be a function defined in an open interval containing [latex]a[\/latex]. The derivative of the function [latex]f(x)[\/latex] at [latex]a[\/latex], denoted by [latex]f^{\\prime}(a)[\/latex], is defined by<\/p>\r\n

[latex]f^{\\prime}(a)=\\underset{x\\to a}{\\lim}\\dfrac{f(x)-f(a)}{x-a}[\/latex]<\/div>\r\n

provided this limit exists.<\/p>\r\n

 <\/p>\r\n

Alternatively, we may also define the derivative of [latex]f(x)[\/latex] at [latex]a[\/latex] as<\/p>\r\n

[latex]f^{\\prime}(a)=\\underset{h\\to 0}{\\lim}\\dfrac{f(a+h)-f(a)}{h}[\/latex]<\/div>\r\n

provided this limit exists.<\/p>\r\n<\/section>\r\n

\r\n

For [latex]f(x)=x^2[\/latex], estimate [latex]f^{\\prime}(3)[\/latex] using the first definition of the derivative.<\/p>\r\n

[reveal-answer q=\"370671\"]Show Answer[\/reveal-answer]
\r\n[hidden-answer a=\"370671\"]<\/p>\r\n

Start by creating a table using values of [latex]x[\/latex] just below [latex]3[\/latex] and just above [latex]3[\/latex].<\/p>\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
[latex]x[\/latex]<\/th>\r\n[latex]\\frac{x^2-9}{x-3}[\/latex]<\/th>\r\n<\/tr>\r\n<\/thead>\r\n
[latex]2.9[\/latex]<\/td>\r\n[latex]5.9[\/latex]<\/td>\r\n<\/tr>\r\n
[latex]2.99[\/latex]<\/td>\r\n[latex]5.99[\/latex]<\/td>\r\n<\/tr>\r\n
[latex]2.999[\/latex]<\/td>\r\n[latex]5.999[\/latex]<\/td>\r\n<\/tr>\r\n
[latex]3.001[\/latex]<\/td>\r\n[latex]6.001[\/latex]<\/td>\r\n<\/tr>\r\n
[latex]3.01[\/latex]<\/td>\r\n[latex]6.01[\/latex]<\/td>\r\n<\/tr>\r\n
[latex]3.1[\/latex]<\/td>\r\n[latex]6.1[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

 <\/p>\r\n

Look at the values in the table. As [latex]x[\/latex] gets closer to [latex]3[\/latex], the values of [latex]\\frac{x^2-9}{x-3}[\/latex] get closer to [latex]6[\/latex].<\/p>\r\n

Based on the table, we can estimate that [latex]f^{\\prime}(3) \\approx 6[\/latex].<\/p>\r\n

[\/hidden-answer]<\/p>\r\n<\/section>\r\n

\r\n

For [latex]f(x)=3x^2-4x+1[\/latex], find [latex]f^{\\prime}(2)[\/latex] by using the second definition.<\/p>\r\n

[reveal-answer q=\"fs-id1169739270484\"]Show Solution[\/reveal-answer]
\r\n[hidden-answer a=\"fs-id1169739270484\"]<\/p>\r\n

Using the second definition, we can substitute two values of the function into the equation.<\/p>\r\n

[latex]\\begin{array}{lllll}f^{\\prime}(2) & =\\underset{h\\to 0}{\\lim}\\frac{f(2+h)-f(2)}{h} & & & \\text{Apply the definition.} \\\\ & =\\underset{h\\to 0}{\\lim}\\frac{(3(2+h)^2-4(2+h)+1)-5}{h} & & & \\begin{array}{l}\\text{Substitute} \\, f(2)=5 \\, \\text{and} \\\\ f(2+h)=3(2+h)^2-4(2+h)+1. \\end{array} \\\\ & =\\underset{h\\to 0}{\\lim}\\frac{3h^2+8h}{h} & & & \\text{Simplify the numerator.} \\\\ & =\\underset{h\\to 0}{\\lim}\\frac{h(3h+8)}{h} & & & \\text{Factor the numerator.} \\\\ & =\\underset{h\\to 0}{\\lim}(3h+8) & & & \\text{Cancel the common factor.} \\\\ & =8 & & & \\text{Evaluate the limit.} \\end{array}[\/latex]<\/div>\r\n

The results are the same whether we use the first or second definition.<\/p>\r\n

[\/hidden-answer]<\/p>\r\n<\/section>\r\n

\r\n

[ohm_question hide_question_numbers=1]162456[\/ohm_question]<\/p>\r\n<\/section>","rendered":"

The Derivative of a Function at a Point<\/h2>\n

The type of limit we compute to find the slope of the tangent line to a function at a point has many applications across various disciplines. These include velocity and acceleration in physics, marginal profit functions in business, and growth rates in biology. This limit occurs so frequently that we give it a special name: the derivative<\/strong>. The process of finding a derivative is called differentiation<\/strong>.<\/p>\n

\n

derivative<\/h3>\n

Let [latex]f(x)[\/latex] be a function defined in an open interval containing [latex]a[\/latex]. The derivative of the function [latex]f(x)[\/latex] at [latex]a[\/latex], denoted by [latex]f^{\\prime}(a)[\/latex], is defined by<\/p>\n

[latex]f^{\\prime}(a)=\\underset{x\\to a}{\\lim}\\dfrac{f(x)-f(a)}{x-a}[\/latex]<\/div>\n

provided this limit exists.<\/p>\n

 <\/p>\n

Alternatively, we may also define the derivative of [latex]f(x)[\/latex] at [latex]a[\/latex] as<\/p>\n

[latex]f^{\\prime}(a)=\\underset{h\\to 0}{\\lim}\\dfrac{f(a+h)-f(a)}{h}[\/latex]<\/div>\n

provided this limit exists.<\/p>\n<\/section>\n

\n

For [latex]f(x)=x^2[\/latex], estimate [latex]f^{\\prime}(3)[\/latex] using the first definition of the derivative.<\/p>\n