For the given curve: y = 5x – 2x3, when x increases at the rate of 2 units/sec, then how fast is the slope of curve changes when x = 3?

To find out how fast the slope of the curve changes, we need to find the derivative of the given function, which represents the slope of the curve at any point x.

The given function is y = 5x – 2x³.

Step 1: Find the derivative of the function. The derivative of y = 5x – 2x³ is y' = 5 - 6x².

Step 2: Substitute x = 3 into the derivative. y'(3) = 5 - 6(3)² = 5 - 6*9 = 5 - 54 = -49.

So, the slope of the curve when x = 3 is -49.

Step 3: Find the rate of change of the slope. The rate of change of the slope is the derivative of the slope with respect to time, which is the derivative of y' = -6x² + 5 with respect to time.

The derivative of y' with respect to time is y'' = -12x(dx/dt).

Step 4: Substitute x = 3 and dx/dt = 2 into the second derivative. y''(3) = -1232 = -72.

So, the slope of the curve is changing at a rate of -72 units/sec when x = 3.