Textbook Question
Using the Alternative Formula for Derivatives
Use the formula
f'(x) = lim (z → x) (f(z) − f(x)) / (z − x)
to find the derivative of the functions in Exercises 23–26.
g(x) = 1 + √x
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 3.3.59
The general polynomial of degree n has the form
P(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₂x² + a₁x + a₀,
where aₙ ≠ 0. Find P'(x).
Verified step by step guidance1
To find the derivative of the polynomial P(x), apply the power rule for differentiation, which states that the derivative of xⁿ is n*xⁿ⁻¹.
Start with the first term aₙxⁿ. The derivative is aₙ * n * xⁿ⁻¹.
Proceed to the next term aₙ₋₁xⁿ⁻¹. The derivative is aₙ₋₁ * (n-1) * xⁿ⁻².
Continue applying the power rule to each term: a₂x² becomes 2*a₂*x, and a₁x becomes a₁.
The constant term a₀ has a derivative of 0, as constants disappear when differentiated. Combine all these derivatives to form P'(x).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Polynomial Function
A polynomial function is an expression consisting of variables and coefficients, structured in terms of powers of a variable. The general form is P(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₁x + a₀, where n is a non-negative integer, and aₙ ≠ 0. Understanding the structure of polynomials is crucial for applying differentiation rules.
Recommended video:
6:04
Introduction to Polynomial Functions
Power Rule for Differentiation
The power rule is a basic principle in calculus used to differentiate functions of the form xⁿ. It states that the derivative of xⁿ is nxⁿ⁻¹. This rule is essential for finding the derivative of each term in a polynomial, allowing us to compute the derivative of the entire polynomial function.
Recommended video:
04:02The Power Rule
Derivative of a Polynomial
The derivative of a polynomial function is obtained by applying the power rule to each term individually. For P(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₁x + a₀, the derivative P'(x) is aₙnxⁿ⁻¹ + aₙ₋₁(n-1)xⁿ⁻² + ... + a₁. This process involves reducing the power of each term by one and multiplying by the original power.
Recommended video:
6:04Introduction to Polynomial Functions
Related Practice
Textbook Question
Finding Derivative Values
In Exercises 67–72, find the value of (f ∘ g)' at the given value of x.
f(u) = u + 1/cos²u, u = g(x) = πx, x = 1/4
Textbook Question
Assume that f'(3) = −1, g'(2) = 5, g(2) = 3, and y = f(g(x)). What is y' at x = 2?
Textbook Question
The edge x of a cube is measured with an error of at most 0.5%. What is the maximum corresponding percentage error in computing the cube’s
a. surface area?
Textbook Question
Find y⁽⁴⁾ = d⁴y/dx⁴ if:
a. y = −2 sin x
Textbook Question
Computer Explorations
Use a CAS to perform the following steps in Exercises 55–62.
a. Plot the equation with the implicit plotter of a CAS. Check to see that the given point P satisfies the equation.
xy³ + tan(x + y) = 1, P(π/4, 0)