Textbook Question
Derivatives in Differential Form
In Exercises 17–28, find dy.
y = (2√x)/(3(1 + √x))
Ch. 3 - Derivatives
Hass15th EditionThomas' CalculusISBN: 9780137616077
Not the one you use?Change textbookSelect a different textbook
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 1
Finding Derivative Functions and Values
Using the definition, calculate the derivatives of the functions in Exercises 1–6. Then find the values of the derivatives as specified.
f(x) = 4 – x²; f′(−3), f′(0), f′(1)
Verified step by step guidance1
Start by recalling the definition of the derivative: f'(x) = lim(h -> 0) [(f(x + h) - f(x)) / h]. This will be used to find the derivative of the function f(x) = 4 - x².
Substitute f(x) = 4 - x² into the definition: f'(x) = lim(h -> 0) [((4 - (x + h)²) - (4 - x²)) / h].
Simplify the expression inside the limit: (4 - (x + h)²) - (4 - x²) = -((x + h)²) + x². Expand (x + h)² to get x² + 2xh + h².
Continue simplifying: -((x² + 2xh + h²) - x²) = -(2xh + h²). The expression becomes: f'(x) = lim(h -> 0) [(-2xh - h²) / h].
Factor out h from the numerator: f'(x) = lim(h -> 0) [-h(2x + h) / h]. Cancel h from numerator and denominator, then evaluate the limit as h approaches 0 to find f'(x) = -2x. Finally, substitute x = -3, 0, and 1 into f'(x) to find f'(-3), f'(0), and f'(1).
Verified video answer for a similar problem:
This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
3mWas this helpful?
Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Definition of Derivative
The derivative of a function at a point is the limit of the average rate of change of the function as the interval approaches zero. Mathematically, it is defined as f'(x) = lim(h→0) [(f(x+h) - f(x))/h]. This concept is essential for understanding how to calculate the instantaneous rate of change of a function.
Recommended video:
05:43
Definition of the Definite Integral
Power Rule for Derivatives
The power rule is a basic derivative rule used to find the derivative of functions in the form of x^n. It states that if f(x) = x^n, then f'(x) = n*x^(n-1). This rule simplifies the process of finding derivatives for polynomial functions, such as f(x) = 4 - x², where the derivative is calculated using this rule.
Recommended video:
Guided course
5:50Power Rules
Evaluating Derivatives at Specific Points
Once the derivative function is found, it can be evaluated at specific points to find the slope of the tangent line at those points. For example, after finding f'(x) for f(x) = 4 - x², you can substitute x = -3, 0, and 1 into f'(x) to find f'(-3), f'(0), and f'(1), respectively, which represent the instantaneous rate of change at these points.
Recommended video:
04:50Critical Points