High School

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State the formula for the derivative (2 options)
a) f'(x) = lim(h->0) [f(x+h) - f(x)]/h
b) f'(x) = lim(h->0) [f(x+h) - f(x-h)]/2h
c) f'(x) = lim(h->0) [f(x) + f(h)]/h
d) f'(x) = lim(h->0) [f(x) - f(h)]/h

Answer :

Final answer:

The correct formulas for the derivative of a function are represented by options (a) and (b), which express the limit definition and the central difference quotient of a derivative, respectively. Options A and B are the correct answers.

Explanation:

The formula for the derivative of a function f with respect to x is an expression that gives the rate at which f changes with x. When seeking the derivative formula, we can use the definition in terms of the limit of the difference quotient.

The correct formulas for the derivative, f'(x), are:

  • f'(x) = lim(h->0) [f(x+h) - f(x)]/h
  • f'(x) = lim(h->0) [f(x+h) - f(x-h)]/2h

Option (a) is the standard definition of a derivative and is correct. Option (b), which uses the central difference quotient, is also a correct method for approximating the derivative, especially in numerical computations. The other options given, (c) and (d), do not correctly represent the formula for a derivative. The final answer, then, is that both options (a) and (b) are correct expressions for finding the derivative of a function.

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