Textbook Question
Calculate the derivative of the following functions.
y = tan(xe^x)
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 63b
Let f(x) = x2 - 6x + 5.
Find the values of x for which the slope of the curve y = f(x) is 2.
Verified step by step guidance1
Step 1: To find the slope of the curve y = f(x), we need to find the derivative of f(x). The function given is f(x) = x^2 - 6x + 5.
Step 2: Differentiate f(x) with respect to x. The derivative, f'(x), represents the slope of the curve at any point x. Use the power rule: if f(x) = ax^n, then f'(x) = n*ax^(n-1).
Step 3: Apply the power rule to each term in f(x). The derivative of x^2 is 2x, the derivative of -6x is -6, and the derivative of a constant (5) is 0. Therefore, f'(x) = 2x - 6.
Step 4: Set the derivative equal to the given slope value. We want the slope to be 2, so set f'(x) = 2x - 6 equal to 2.
Step 5: Solve the equation 2x - 6 = 2 for x. This will give you the x-values where the slope of the curve is 2.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Derivative
The derivative of a function measures the rate at which the function's value changes as its input changes. It is often interpreted as the slope of the tangent line to the curve at a given point. For the function f(x) = x² - 6x + 5, finding the derivative f'(x) will allow us to determine the slope of the curve at any point x.
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Derivatives
Slope of the Curve
The slope of the curve at a specific point is given by the value of the derivative at that point. In this problem, we are interested in finding the values of x where the slope of the curve, represented by f'(x), equals 2. This involves setting the derivative equal to 2 and solving for x.
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Quadratic Functions
A quadratic function is a polynomial function of degree two, typically expressed in the form f(x) = ax² + bx + c. The graph of a quadratic function is a parabola. In this case, the function f(x) = x² - 6x + 5 is a quadratic function, and understanding its properties, such as its vertex and axis of symmetry, can provide insights into its behavior and the solutions to the slope problem.
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Related Practice
Textbook Question
Calculate the derivative of the following functions.
y = (e^x / x+1)⁸
Textbook Question
A line perpendicular to another line or to a tangent line is often called a normal line. Find an equation of the line perpendicular to the line that is tangent to the following curves at the given point P.
y=3x−4; P(1, −1)
Textbook Question
A line perpendicular to another line or to a tangent line is often called a normal line. Find an equation of the line perpendicular to the line that is tangent to the following curves at the given point P.
y = 2/x; P(1, 2)
Textbook Question
Use a graphing utility to graph the curve and the tangent line on the same set of axes.
y = 2x2 / (3x - 1); a = 1
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Textbook Question
Let f(x) = x2 - 6x + 5.
Find the values of x for which the slope of the curve y = f(x) is 0.
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