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 Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- 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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Video duration:
2mKey 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.
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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.
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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.
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6:04Introduction to Polynomial Functions
Related Practice
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.
g(x) = 8 / x², (2, 2)
Textbook Question
If the original 24 m edge length x of a cube decreases at the rate of 5 m/min, when x = 3 m at what rate does the cube’s
a. surface area change?
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
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)