Problem 74 - Algebra and Trigonometry

Question

\(71-76\) m simplify the expression. (This type of expression arises in calculus when using the "quotient rule.") $$ \frac{\left(1-x^{2}\right)^{1 / 2}+x^{2}\left(1-x^{2}\right)^{-1 / 2}}{1-x^{2}} $$

Step-by-Step Solution

Answer

The expression simplifies to \( (1-x^2)^{-3/2} \).

Step 1: Recognize the Problem Type

This expression involves square roots and a function in the denominator. Simplifying expressions like this is often seen in calculus problems when using rules like the quotient rule.

Step 2: Expand the Expression

The given expression is \( \frac{(1-x^2)^{1/2} + x^2(1-x^2)^{-1/2}}{1-x^2} \). A common technique is to combine terms under the square root for a clearer form.

Step 3: Simplify the Numerator

Rewriting the numerator \((1-x^2)^{1/2} + x^2(1-x^2)^{-1/2}\), we combine these under a common denominator: \(\frac{(1-x^2) + x^2}{(1-x^2)^{1/2}}\). This simplifies to: \( \frac{1}{(1-x^2)^{1/2}}\).

Step 4: Simplify the Overall Expression

Now divide the simplified numerator \( \frac{1}{(1-x^2)^{1/2}} \) by the denominator \( 1-x^2 \). This simplifies to \( \frac{1}{(1-x^2)^{3/2}} \).

Step 5: Review the Simplified Expression

The expression is now fully simplified as \( (1-x^2)^{-3/2} \). Ensure no further simplifications are available.

Key Concepts

Simplifying Expressions

Simplifying expressions is all about making complex mathematical expressions easier to work with. It involves reducing an expression to its most basic form. The process makes calculation and understanding more straightforward. In calculus, simplification often involves:

• Combining like terms.
• Breaking down complex fractions.
• Using algebraic identities to condense expressions.

For this particular exercise, the expression was challenging due to the presence of square roots and fractional exponents. The key was to merge terms under the square root and common denominators. This reduces unnecessary complexity, making it more manageable to solve. Remember, simplifying expressions is a fundamental skill that aids in other calculus techniques, like differentiation and integration.

Quotient Rule

The quotient rule is a method used in calculus to find the derivative of a quotient of two functions. Essentially, if you have a function that is the division of two separate functions, you'll use this rule. The quotient rule can be given by:\[\frac{d}{dx}\left(\frac{u(x)}{v(x)}\right) = \frac{u'(x)v(x) - u(x)v'(x)}{(v(x))^2}\]Where:

• v(x) are functions of x.
• u'(x) and v'(x) are derivatives of u(x) and v(x) respectively.

Understanding how to simplify such expressions, like in our exercise, can be essential before applying the quotient rule. Simplifying might reveal easier derivatives where otherwise it would be complicated. In other words, breaking down the fraction to its basic components often helps to see the structure more clearly.

Square Roots

Square roots challenge students because they involve finding a number that gives the original number when squared. In expressions, radical symbols or fractional exponents represent square roots. For example, \((1-x^2)^{1/2}\) and \((1-x^2)^{-1/2}\) signify the presence of square roots. When working with such expressions:

• Seek to find common bases to simplify them.
• Recognize that \((a)^{1/2}\) is the same as \(\sqrt{a}\).
• Understand that negative exponents, like \((1-x^2)^{-1/2}\), reflect reciprocal square roots.

Effectively managing square roots allows easier manipulation of expressions in calculus, paving the way for deriving and integrating functions effectively. Comprehending the nature of square roots and their simplification is vital in solving calculus problems like the one detailed in the exercise.

Function Simplification

Function simplification in calculus refers to the process of changing a complex function into a simpler form. Simplified forms make functions easier to differentiate, integrate, or analyze. Techniques for function simplification often involve:

• Combining terms with common variables and operations.
• Eliminating complex fractions by rationalizing denominators.
• Using known algebraic identities and calculus properties to reduce terms.

In the given problem, such simplification facilitates the use of further calculus techniques. When dealing with expressions like \( \frac{1}{(1-x^2)^{3/2}}\), breaking down and simplifying before applying rules like differentiation can prevent errors. Always look for patterns or consistent terms when simplifying functions, as this is fundamental to mastering calculus.