determine which of the following derivative notations are correct, and which are not, for the given function…

determine which of the following derivative notations are correct, and which are not, for the given function y = f(x). drag each of the above items into the appropriate area below, depending on whether it represents correct or incorrect notation for the derivative of the given function. f(x)=8x³ + 7x + 4

determine which of the following derivative notations are correct, and which are not, for the given function y = f(x). drag each of the above items into the appropriate area below, depending on whether it represents correct or incorrect notation for the derivative of the given function. f(x)=8x³ + 7x + 4

Answer

Explanation:

Step1: Recall derivative notations

For a function $y = f(x)$, common derivative notations include $\frac{dy}{dx}$, $D_x(f(x))$, $f^{\prime}(x)$.

Step2: Analyze each notation

  • $\frac{dy}{dx}$: Correct as it represents the derivative of $y$ with respect to $x$ for $y = f(x)$.
  • $D_x(8x^{3}+7x + 4)$: Correct, it is the operator - based notation for the derivative of the function $8x^{3}+7x + 4$ with respect to $x$.
  • $\frac{dy}{dx}(8x^{3}+7x + 4)$: Incorrect. $\frac{dy}{dx}$ already represents the derivative operation, and writing it this way is a misuse of notation.
  • $f^{\prime}(x)$: Correct, it is the prime - notation for the derivative of the function $f(x)$.

Answer:

Incorrect Derivative Notation: $\frac{dy}{dx}(8x^{3}+7x + 4)$ Correct Derivative Notation: $\frac{dy}{dx}$, $D_x(8x^{3}+7x + 4)$, $f^{\prime}(x)$