Textbook Question
Finding derivatives from a table Find the values of the following derivatives using the table. <IMAGE>
a. d/dx (f(x)+2g(x)) |x=3
3. Techniques of Differentiation
Basic Rules of Differentiation
4:01 minutesProblem 3.3.56
Textbook Question
Find the derivative of the following functions by first expanding or simplifying the expression. Simplify your answers.
y = (x2 - 2ax + a2) / (x - a); a is a constant.
Verified step by step guidance1
Step 1: Recognize that the function y = \frac{x^2 - 2ax + a^2}{x - a} is a rational function, which can be simplified by performing polynomial long division or by recognizing it as a special form.
Step 2: Notice that the numerator x^2 - 2ax + a^2 can be rewritten as (x - a)^2 by recognizing it as a perfect square trinomial.
Step 3: Simplify the expression by canceling the common factor (x - a) in the numerator and the denominator, resulting in y = x - a.
Step 4: Differentiate the simplified function y = x - a with respect to x. Since a is a constant, the derivative of a constant is zero.
Step 5: Apply the basic differentiation rule: the derivative of x with respect to x is 1. Therefore, the derivative of y = x - a is y' = 1.
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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 how the function's output value changes as its input value changes. It is defined as the limit of the average rate of change of the function over an interval as the interval approaches zero. In calculus, the derivative is often denoted as f'(x) or dy/dx, and it provides critical information about the function's behavior, such as its slope and points of tangency.
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Simplification of Expressions
Simplifying an expression involves rewriting it in a more manageable or understandable form, often by combining like terms, factoring, or reducing fractions. In the context of calculus, simplifying an expression before taking the derivative can make the differentiation process easier and help avoid errors. For example, factoring polynomials or canceling common terms can lead to a clearer function to differentiate.
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Quotient Rule
The quotient rule is a method for finding the derivative of a function that is the ratio of two other functions. If y = u/v, where u and v are functions of x, the derivative is given by y' = (v * u' - u * v') / v^2. This rule is essential when dealing with functions that are expressed as fractions, as it allows for the correct application of differentiation to both the numerator and denominator.
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