#### Question

For the curve y = 5x – 2x^{3}, if x increases at the rate of 2 units/sec, then find the rate of change of the slope of the curve when x = 3

#### Solution

The given curve is y = 5x – 2x^{3} and `dx/dt = 2` units/sec

y = 5x – 2x^{3}

Differentiating both sides w.r.t x, we get

Slope of the curve = `dy/dx = 5 - 6x^2`

Differentiating both sides w.r.t *t*, we get

`=> d/dt (dy/dx) = 0 - 12x dx/dt`

`=> d/dt (dy/dx)_(x = 3) = 0 - 12 xx 3 xx2 = -72 "units/sec"`

Thus, the slope of the curve is decreasing at the rate of 72 units/sec when x = 3

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Solution For the Curve Y = 5x – 2x3, If X Increases at the Rate of 2 Units/Sec, Then Find the Rate of Change of the Slope of the Curve When X = 3 Concept: Procedure to Form a Differential Equation that Will Represent a Given Family of Curves.