Textbook Question
Find the derivative function f' for the following functions f.
f(x) = 2/3x+1; a= -1
Ch. 3 - Derivatives
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 3, Problem 40b
Find an equation of the line tangent to the graph of f at (a, f(a)) for the given value of a.
f(x) = 1/x; a= -5
Verified step by step guidance1
Step 1: Understand that the equation of a tangent line to the graph of a function at a point (a, f(a)) is given by the formula y - f(a) = f'(a)(x - a), where f'(a) is the derivative of the function evaluated at x = a.
Step 2: Find the derivative of the function f(x) = 1/x. The derivative, f'(x), can be found using the power rule. Rewrite f(x) as x^(-1) and differentiate to get f'(x) = -1/x^2.
Step 3: Evaluate the derivative at the given point a = -5. Substitute x = -5 into f'(x) to find f'(-5) = -1/(-5)^2.
Step 4: Calculate f(a) by substituting a = -5 into the original function f(x) = 1/x. This gives f(-5) = 1/(-5).
Step 5: Substitute f(a) and f'(a) into the tangent line equation y - f(a) = f'(a)(x - a) to find the equation of the tangent line at the point (a, f(a)).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Tangent Line
A tangent line to a curve at a given point is a straight line that touches the curve at that point without crossing it. The slope of the tangent line represents the instantaneous rate of change of the function at that point, which is determined by the derivative of the function.
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Derivative
The derivative of a function at a point quantifies how the function's output changes as its input changes. It is calculated as the limit of the average rate of change of the function as the interval approaches zero. For the function f(x) = 1/x, the derivative can be found using the power rule or quotient rule.
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Point-Slope Form
The point-slope form of a linear equation is used to express the equation of a line given a point on the line and its slope. It is written as y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the slope. This form is particularly useful for finding the equation of the tangent line once the slope and point are known.
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Related Practice
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