Question

If y = 7x − x³ and x increases at the rate of 4 units per second, how fast is the slope of the curve changing when x = 2?

Answer 1

$$\begin{gathered} \mathrm{Here}, \\ y=7x-x^3 \\ \Rightarrow \frac{dy}{dx}=7x-x^3 \\ \mathrm{Let}s\mathrm{be}\mathrm{the}\mathrm{slope}.\mathrm{Then}, \\ s=7-3x^2 \\ \Rightarrow \frac{ds}{dt}=-6x\frac{dx}{dt} \\ \Rightarrow \frac{ds}{dt}=-6\left(4\right)\left(2\right)\left[\because x=2\mathrm{and}\frac{dx}{dt}=4\mathrm{units}/\sec \right] \\ \Rightarrow \frac{ds}{dt}=-48 \end{gathered}$$