Question

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

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