If `y=7x-x^3` and `x` increases at the rate of `4` units per second, how fast is the slope of the curve changing when `x=2` ?

Explanation

`y=7 x-x^{3} `


`Rightarrow frac{d y}{d x}=7 x-x^{3}`



Let `s` be the slope. Then



`s=7-3 x^{2} `


`Rightarrow frac{d s}{d t}=-6 x frac{d x}{d t} `


`Rightarrow frac{d s}{d t}=-6(4)(2)[because x=2 text { and } frac{d x}{d t}=4 text { units } / {sec}] `


...