Math Problem Statement
difference quotient of f(x) = (x-1)/(x+2)
Solution
To find the difference quotient of the function , we'll use the formula for the difference quotient:
Step-by-step process:
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Find : Substitute into :
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Compute : Now, subtract from :
To subtract these fractions, first find a common denominator. The common denominator is . So, we have:
Expanding both numerators: Simplifying:
Similarly, expand : Simplifying:
Now, subtract the second numerator from the first:
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Simplify the difference quotient expression: Now, we can express as:
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Divide by : Finally, divide by : Simplifying further:
Thus, the difference quotient of is:
Would you like more details on any step or a further breakdown?
Here are 5 related questions for further practice:
- What is the derivative of ?
- How does the difference quotient relate to the derivative?
- What happens to the difference quotient as ?
- How do you simplify difference quotients with more complex functions?
- What is the difference quotient of ?
Tip: The difference quotient is essentially the slope of the secant line between two points on the curve, and it approaches the derivative as the distance between the points shrinks (as ).
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Math Problem Analysis
Mathematical Concepts
Calculus
Difference Quotient
Algebra
Rational Functions
Formulas
Difference Quotient: (f(x+h) - f(x)) / h
Theorems
Definition of Derivative
Properties of Rational Functions
Suitable Grade Level
Grades 11-12 (Pre-Calculus/Calculus)