Calculus: Derivatives of Functions
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Questions and Answers

What is the derivative of the function $\frac{1}{x^2 + 1}$?

  • $\frac{-1}{(x^2 + 1)^2}$
  • $\frac{-2x}{(x^2 + 1)^2}$ (correct)
  • $\frac{1}{(x^2 + 1)^2}$
  • $\frac{2x}{(x^2 + 1)^2}$

    • What is the derivative of the function $\frac{e^x}{x^2}$?

  • $\frac{e^x(x - 2)}{x^2}$
  • $\frac{e^x(x^2 - 2x)}{x^4}$
  • $\frac{e^x(x^2 + 2)}{x^4}$
  • $\frac{e^x(x - 2)}{x^4}$ (correct)

    • Which of the following is the derivative of the reciprocal of sin(x)?

  • $\frac{-\cos(x)}{\sin^2(x)}$ (correct)
  • $\frac{-1}{\cos^2(x)}$
  • $\frac{\cos(x)}{\sin^2(x)}$
  • $\frac{1}{\cos^2(x)}$

    • What is the derivative of $\frac{x^3}{x^2 + 1}$?

    <p>$\frac{x^4 + 3x^2}{(x^2 + 1)^2}$ (B)</p> Signup and view all the answers

    If $f(x) = \frac{1}{x^3 + 2}$, what is the derivative of $f(x)$?

    <p>$\frac{-3x^2}{(x^3 + 2)^2}$ (B)</p> Signup and view all the answers

    Which of the following is the derivative of $\frac{x^2 + 1}{x}$?

    <p>$\frac{x^2 - 1}{x^2}$ (A)</p> Signup and view all the answers

    What is the derivative of $\frac{\ln(x)}{x}$?

    <p>$\frac{1 - \ln(x)}{x^2}$ (B)</p> Signup and view all the answers

    What is the derivative of the function $\frac{1}{\sqrt{x}}$?

    <p>$\frac{-1}{2x\sqrt{x}}$ (C)</p> Signup and view all the answers

    Which of the following is the derivative of $\frac{cos(x)}{sin(x)}$?

    <p>$\frac{-1}{\sin^2(x)}$ (C)</p> Signup and view all the answers

    Find the derivative of the function $\frac{x}{x^2 + 1}$

    <p>$\frac{1 - x^2}{(x^2 + 1)^2}$ (D)</p> Signup and view all the answers

    Flashcards

    Derivative of Reciprocal

    The derivative of the reciprocal of a function is given by: -f'(x)/[f(x)]².

    Limit Calculation

    To find the derivative, use the limit: lim(h→0)[(1/f(x+h)) - (1/f(x))]/h.

    Common Denominator

    Use common denominators to simplify: (f(x) - f(x+h))/(h) * (1/(f(x+h)f(x))).

    Two-part Limit

    Split the limit into: lim(h→0)[(f(x)-f(x+h))/h] and lim(h→0)[1/(f(x+h)f(x))].

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    Quotient Rule

    The derivative of a quotient p(x)/q(x) is (p'(x)q(x) - p(x)q'(x))/[q(x)]².

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    Product of Functions

    The quotient can be seen as the product of p(x) and the reciprocal of q(x).

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    Reciprocal Derivative Rule

    The derivative of 1/q(x) is -q'(x)/[q(x)]².

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    Derivative Example 1

    d/dx of (1/cos(x)) = sin(x)/cos²(x).

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    Derivative Example 2

    d/dx of (sin(x)/cos(x)) simplifies to 1/cos²(x).

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    Product Rule Application

    In derivative of (e^x/x), use product rule for adjustment: (xe^x - e^x)/x².

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    Study Notes

    Derivative of the Reciprocal of a Function

    • The derivative of the reciprocal of a function $f(x)$ is given by:
      • $\frac{d}{dx} \left(\frac{1}{f(x)}\right) = -\frac{f'(x)}{[f(x)]^2}$
    • Derivation involves calculating the limit of the incremental ratio:
      • $\lim_{h \to 0}\frac{\frac{1}{f(x+h)} - \frac{1}{f(x)}}{h}$
    • Simplifying by common denominators yields:
      • $\lim_{h \to 0} \frac{f(x) - f(x+h)}{h} * \frac{1}{f(x+h)f(x)}$
    • This limit splits into two parts:
      • $\lim_{h \to 0} \frac{f(x) - f(x+h)}{h}$ with a limit of $-f'(x)$ as $h$ approaches $0$
      • $\lim_{h \to 0} \frac{1}{f(x+h)f(x)}$ with a limit of $\frac{1}{[f(x)]^2}$ as $h$ approaches $0$
    • Multiplying the two limits produces the original formula.

    Derivative of the Quotient of Two Functions

    • The derivative of the quotient of two functions $p(x)$ and $q(x)$ is given by:
      • $\frac{d}{dx} \left(\frac{p(x)}{q(x)}\right)= \frac{p'(x)q(x) - p(x)q'(x)}{[q(x)]^2}$
    • This formula derives from viewing the quotient as a product:
      • $\frac{p(x)}{q(x)} = p(x) * \frac{1}{q(x)}$
    • Using the product rule and the reciprocal rule, the derivative of the quotient is obtained.

    Examples

    • Reciprocal Examples:
      • $\frac{d}{dx} \left(\frac{1}{\cos(x)}\right) = \frac{\sin(x)}{\cos^2(x)}$
      • $\frac{d}{dx} \left(\frac{1}{\ln(x)}\right) = -\frac{1}{x\ln^2(x)}$
    • Quotient Examples:
      • $\frac{d}{dx} \left(\frac{\sin(x)}{\cos(x)}\right) = \frac{\cos^2(x) + \sin^2(x)}{\cos^2(x)} = \frac{1}{\cos^2(x)}$
      • $\frac{d}{dx} \left(\frac{e^x}{x}\right) = \frac{xe^x - e^x}{x^2}$

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    Description

    Explore the derivatives of the reciprocal and quotient of functions in this calculus quiz. Understand how to apply the derivative formulas and simplify expressions to find the limits. Perfect for students looking to enhance their calculus skills.

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