Given a curve $y = 7x - x^3$ and $x$ increase at the rate of 2 units per second. The rate at which the slope of the curve is changing, when $x = 5$ is:

$-60 \text{ units/s}$

The correct answer is Option (1) → $-60 \text{ units/s}$ ##

Given,

$ y = 7x - x^3 $

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

$ \frac{dy}{dx} = 7 - 3x^2 $

Thus, slope,  $ s = 7 - 3x^2 $

Now, differentiating w.r.t. '$t$', we get

$ \frac{ds}{dt} = -6x \frac{dx}{dt} $

$ ∴\left. \frac{ds}{dt} \right|_{x = 5} = -6 \times 5 \times 2 $  $ [∵\frac{dx}{dt} = 2 \text{ units/s (given)}] $

$ = -60 \text{ units/s} $