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Закон руху точки по прямій задається формулою $s(t) = 14t^2 - 2t$ , де $t$ - час (в секундах), $s(t)$ - відхилення точки в момент часу $t$ (в метрах) від початкового положення. Знайди миттєву швидкість руху точки

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Find derivatives using the quotient rule II

Full solution

Q. Закон руху точки по прямій задається формулою

s(t) = 14t^2 - 2t

, де

t

- час (в секундах),

s(t)

- відхилення точки в момент часу

t

(в метрах) від початкового положення. Знайди миттєву швидкість руху точки

Identify function: Identify the function to differentiate. $\newline$ The function that describes the motion of the point is $s(t) = 14t^2 - 2t$ . To find the instantaneous velocity, we need to find the derivative of this function with respect to time $t$ .
Differentiate function: Differentiate the function with respect to $t$ . The derivative of $s(t)$ with respect to $t$ is $s'(t)$ , which represents the instantaneous velocity. To differentiate $s(t) = 14t^2 - 2t$ , we apply the power rule, which states that the derivative of $t^n$ is $n\cdot t^{(n-1)}$ .
Calculate derivative: Calculate the derivative using the power rule. $\newline$ The derivative of the first term, $14t^2$ , is $2\times 14\times t^{2-1} = 28t$ . $\newline$ The derivative of the second term, $-2t$ , is $-2\times 1\times t^{1-1} = -2$ . $\newline$ So, $s'(t) = 28t - 2$ .
Check for errors: Check for any mathematical errors. $\newline$ There are no mathematical errors in the differentiation process.
Write final answer: Write the final answer. $\newline$ The instantaneous velocity of the point at time $t$ is given by the derivative $s'(t) = 28t - 2$ .

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