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For the following equation, what is the instantaneous rate of change at
x=-2?

f(x)=-x^(3)-2x
Answer:

For the following equation, what is the instantaneous rate of change at $x=-2 ?$ $\newline$ $f(x)=-x^{3}-2 x$ $\newline$ Answer:

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Find derivatives using logarithmic differentiation

Full solution

Q. For the following equation, what is the instantaneous rate of change at

x=-2 ?

\newline

f(x)=-x^{3}-2 x

\newline

Answer:

Calculate Derivative: To find the instantaneous rate of change of the function at a specific point, we need to calculate the derivative of the function. The derivative of a function at a point gives us the slope of the tangent line at that point, which is the instantaneous rate of change.
Apply Power Rule: The function given is $f(x) = -x^3 - 2x$ . We will find the derivative $f'(x)$ using the power rule. The power rule states that the derivative of $x^n$ is $n\cdot x^{(n-1)}$ .
Evaluate at $x = -2$ : Applying the power rule to each term in the function: $\newline$ The derivative of $-x^3$ is $-3x^2$ (using the power rule). $\newline$ The derivative of $-2x$ is $-2$ (since the derivative of $x$ is $1$ and the constant multiple rule applies). $\newline$ So, $f'(x) = -3x^2 - 2$ .
Substitute and Calculate: Now we need to evaluate the derivative at $x = -2$ to find the instantaneous rate of change at that point. $\newline$ Substitute $x = -2$ into $f'(x)$ : $\newline$ $f'(-2) = -3(-2)^2 - 2$ .
Substitute and Calculate: Now we need to evaluate the derivative at $x = -2$ to find the instantaneous rate of change at that point. $\newline$ Substitute $x = -2$ into $f'(x)$ : $\newline$ $f'(-2) = -3(-2)^2 - 2$ .Calculate the value: $\newline$ $f'(-2) = -3(4) - 2 = -12 - 2 = -14$ .

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