Given as equation of curve y = 7x – x3 and x increases at the rate of 4 units per second.
As to find how fast is the slope of the curve changing when x = 2
The equation of curve is y = 7x – x3
Differentiate the above equation with respect to x, we get slope of the curve
dy/dx = d(7x - x3)/dx
Suppose m be the slope of the given curve then the above equation becomes,
m = 7 - 3x2 ...(ii)
Given x increases at the rate of 4 units per second, therefore
dx/dt = 4 units/sec ...(iii)
Differentiate the equation of slope that is equation (ii)
When x = 2, equation (iv) becomes
dm/dt = (-6x) x (4) = -6 x 2 x 4 = - 48
The slope cannot be negative,
Thus, the slope of the curve is changing at the rate of 48 units/sec when x = 2