Maths · Applications of derivatives: Rate of change of quantities, monotonic-Increasing and decreasing functions, Maxima and minima of functions of one variable
If and increases at the rate of 5 units per second, the rate of change of slope
If \( y=6 x-x^{3} \) and \( x \) increases at the rate of 5 units per second, the rate of change of slope when \( x=3 \) is
- A. -90 units/sec
- B. 90 units/ sec
- C. 180 units/sec
- D. -180 units/sec
Step-by-step solution
The slope of y = 6x - x^3 is dy/dx = 6 - 3x^2. The rate of change of slope is d/dt(dy/dx) = d^2y/dx^2 * dx/dt = (-6x) * 5. At x=3, this is (-18)*5 = -90 units/sec.
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