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Question
If y = 7x − x3 and x increases at the rate of 4 units per second, how fast is the slope of the curve changing when x = 2?
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Solution
\[\text {Here }, \]
\[y = 7x - x^3 \]
\[ \Rightarrow \frac{dy}{dx} = 7x - x^3 \]
\[\text { Let s be the slope . Then }, \]
\[s = 7 - 3 x^2 \]
\[ \Rightarrow \frac{ds}{dt} = - 6x\frac{dx}{dt}\]
\[ \Rightarrow \frac{ds}{dt} = - 6\left( 4 \right)\left( 2 \right) \left[ \because x = 2 \text { and } \frac{dx}{dt} = 4 \text { units }/\sec \right]\]
\[ \Rightarrow \frac{ds}{dt} = - 48\]
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