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

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