Rules for Differentiation
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Questions and Answers

What is the derivative of $x^n?$

  • $x^{n+1}$
  • $nx^{n-2}$
  • $nx^{n-1}$ (correct)
  • $n^2x^n$

    • What is the derivative of a constant?

  • $k$
  • $0$ (correct)
  • $k^2$
  • $1$

    • What is the derivative of $k * f(x)$?

  • $k^2 \* f(x)$
  • $f(x)$
  • $k \* f'(x)$ (correct)
  • $f'(x)$

    • What is the derivative of f(x) + g(x)?

    <p>$\frac{d}{dx} [f(x)] + \frac{d}{dx} [g(x)]$</p> Signup and view all the answers

    What is the notation for the derivative of y with respect to x?

    <p>All of the above</p> Signup and view all the answers

    Study Notes

    Rules for Differentiation

    • Determining the derivative of a function from first principles requires a long calculation and is prone to mistakes.
    • The general rule for differentiation is: d/dx [x^n] = nx^(n-1), where n is a real number and n is not equal to 0.

    Derivative Rules

    • Derivative of a constant: d/dx [k] = 0.
    • Derivative of a constant multiplied by a function: d/dx [k * f(x)] = k * d/dx [f(x)].
    • Derivative of a sum: d/dx [f(x) + g(x)] = d/dx [f(x)] + d/dx [g(x)].
    • Derivative of a difference: d/dx [f(x) - g(x)] = d/dx [f(x)] - d/dx [g(x)].

    When to Use the Rules for Differentiation

    • Use the rules for differentiation when the question does not specify how to determine the derivative.
    • Use first principles if specifically requested or if asked to differentiate using the definition of a derivative.

    Notation

    • Alternative notations for the derivative include: f'(x), y', dy/dx, df/dx, d/dx [f(x)], Df(x), and D_xy.
    • D and d/dx are called differential operators because they indicate the operation of differentiation.
    • dy/dx means y differentiated with respect to x.

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    Description

    Learn the rules for differentiation, including the general rule for finding the derivative of a function, making it easier to calculate derivatives.

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