Mathematics

Differentiation and Algebraic Fractions

33 Questions

Differentiation combined with algebraic fractions involves solving implicit functions and evaluating complex mathematical ratios. Test items require calculating second derivatives, inverse trigonometric functions, and variable fraction equivalencies. This specialized mathematics topic appears regularly in high level aptitude screenings.

Implicit function derivativesAlgebraic fraction ratiosSecond derivative calculationInverse trigonometric differentiationVariable fraction equivalencies

Differentiation and Algebraic Fractions Questions

Multiple choice maths 5-digit numbers expanded form introduction to numbers and number systems numbers in general form

Prove that $\dfrac{a^{-1}}{(a^{-1}+b^{-1})}$ is equal to $\dfrac{b}{(a+b)}$

  1. True

  2. False
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation
$\cfrac{{ a }^{ -1 }}{{ a }^{ -1 }+{ b }^{ -1 }}\Leftrightarrow \cfrac{{ a }^{ -1 }}{\cfrac{1}{a}+\cfrac{1}{b}}$
$\Rightarrow$ $\cfrac{{ a }^{ -1 }}{\cfrac{b+a}{a.b}}$
$\Rightarrow$ $\cfrac{a.b}{a(b+a)}$
$\Rightarrow$ $\cfrac{b}{b+a}$
$\Rightarrow$ $\cfrac{b}{a+b}$
$\therefore$ $\cfrac{{ a }^{ -1 }}{{ a }^{ -1 }+{ b }^{ -1 }}=\cfrac{b}{a+b}$
Multiple choice

The sum of 1/ \frac{1}{x+y}x + y and \frac{1}{x-y} is

  1. <latex>\frac{2x}{x^2-y^2}</latex>

  2. <latex>\frac{2}{2x}</latex>

  3. <latex>\frac{2y}{x^2-y^2}</latex>

  4. <latex>\frac{-2x}{x^2-y^2}</latex>
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The sum is 1/(x + y) + 1/(x - y). Find a common denominator: (x - y)/((x + y)(x - y)) + (x + y)/((x + y)(x - y)) = (x - y + x + y) / (x^2 - y^2) = 2x / (x^2 - y^2).