For the curve y=5x-2x^{3},if x increases at t | vedclass

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(A) The slope of the curve is given by $m = \frac{dy}{dx} = 5 - 6x^{2}$.To find the rate of change of the slope with respect to time $t$,we differenti

Vedclass · Accepted Answer

-$72$ units/sec

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(A) The slope of the curve is given by $m = \frac{dy}{dx} = 5 - 6x^{2}$.To find the rate of change of the slope with respect to time $t$,we differentiate $m$ with respect to $t$ using the chain rule:$\frac{dm}{dt} = \frac{d}{dt}(5 - 6x^{2}) = -12x \cdot \frac{dx}{dt}$.Given that $x = 3$ and $\frac{dx}{dt} = 2 	ext{ units/sec}$,we substitute these values into the equation:$\frac{dm}{dt} = -12(3)(2) = -72 	ext{ units/sec}$.Thus,the slope of the curve is changing at the rate of $-72 	ext{ units/sec}$,which means it is decreasing at a rate of $72 	ext{ units/sec}$.

For the curve $y=5x-2x^{3}$,if $x$ increases at the rate of $2 \text{ units/sec}$,then how fast is the slope of the curve changing when $x=3$?

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