Textbook Question
In Exercises 11–18, find the slope of the function’s graph at the given point. Then find an equation for the line tangent to the graph there.
f(x) = √(x + 1), (8, 3)
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Ch. 3 - Derivatives
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 3, Problem 3.5.17
Derivatives
In Exercises 1–18, find dy/dx.
f(x) = x³ sin x cos x
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Identify the function f(x) = x³ sin(x) cos(x) and recognize that it is a product of three functions: u(x) = x³, v(x) = sin(x), and w(x) = cos(x).
Apply the product rule for derivatives, which states that if you have a product of functions u(x), v(x), and w(x), then the derivative is given by: (uvw)' = u'vw + uv'w + uvw'.
Calculate the derivative of each individual function: u'(x) = 3x², v'(x) = cos(x), and w'(x) = -sin(x).
Substitute these derivatives into the product rule formula: dy/dx = (3x²)(sin(x))(cos(x)) + (x³)(cos(x))(cos(x)) + (x³)(sin(x))(-sin(x)).
Simplify the expression by combining like terms and using trigonometric identities if necessary to express the derivative in its simplest form.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Derivatives
A derivative represents the rate of change of a function with respect to its variable. It is a fundamental concept in calculus that provides information about the slope of the tangent line to the curve of the function at any given point. The notation dy/dx indicates the derivative of y with respect to x, and it can be computed using various rules such as the power rule, product rule, and chain rule.
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Derivatives
Product Rule
The product rule is a formula used to find the derivative of the product of two functions. If u(x) and v(x) are two differentiable functions, the product rule states that the derivative of their product is given by d(uv)/dx = u'v + uv', where u' and v' are the derivatives of u and v, respectively. This rule is essential for differentiating functions that are products of simpler functions, such as the given function f(x) = x³ sin x cos x.
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05:18The Product Rule
Trigonometric Functions
Trigonometric functions, such as sine (sin) and cosine (cos), are fundamental in calculus, especially when dealing with derivatives. These functions have specific derivatives: the derivative of sin(x) is cos(x), and the derivative of cos(x) is -sin(x). Understanding these derivatives is crucial when applying the product rule to functions that involve trigonometric components, as seen in the function f(x) = x³ sin x cos x.
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6:04Introduction to Trigonometric Functions
Related Practice
Textbook Question
Tangent line to y = √x Does any tangent line to the curve y = √x cross the x-axis at x = −1? If so, find an equation for the line and the point of tangency. If not, why not?
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Textbook Question
Derivatives
In Exercises 27–32, find dp/dq.
p = (q sin q) / (q² − 1)
244
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Textbook Question
Finding Derivative Values
In Exercises 67–72, find the value of (f ∘ g)' at the given value of x.
f(u) = 1 − (1/u), u = g(x) = (1 / (1 − x)), x = −1
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Textbook Question
Tangent Lines
In Exercises 35–38, graph the curves over the given intervals, together with their tangent lines at the given values of x. Label each curve and tangent line with its equation.
y = sin x, −3π/2 ≤ x ≤ 2π
x = −π, 0, 3π/2
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Textbook Question
In Exercises 5–10, find an equation for the tangent line to the curve at the given point. Then sketch the curve and tangent line together.
y = 4 − x², (−1, 3)
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