For the curve `y=5x-2x^3` , if `x` increases at the rate of 2 units/sec, then how fast is the slope of the curve changing when `x=3?`

Explanation

The given curve is `y=5 x-2 x^{3}` and `frac{d x}{d t}=2` units/sec `y=5 x-2 x^{3}`


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


Slope of the curve `=frac{d y}{d x}=5-6 x^{2}`


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



`Rightarrow frac{d}{d t}(frac{d y}{d x})=0-12 x frac{d x}{d t} `


`Rightarrow frac{d}{d t}(frac{d y}{d x})_{x=3}=0-12 times 3 times 2=-72 {units} / {sec}`



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