Step 1 (Formula / Governing Law): [1/2 Mark]

The slope of the curve $y = f(x)$ is given by $m = \frac{dy}{dx}$. We are required to find the rate of change of the slope with respect to time $t$, which is $\frac{dm}{dt}$. By the chain rule:

$$\frac{dm}{dt} = \frac{d}{dt}\left(\frac{dy}{dx}\right) = \frac{d}{dx}\left(\frac{dy}{dx}\right) \cdot \frac{dx}{dt} = \frac{d^2y}{dx^2} \cdot \frac{dx}{dt}$$

Step 2 (Step-by-Step Working): [1/4 Mark]

• Given curve: $y = 7x - x^3$
• First derivative (slope $m$): $\frac{dy}{dx} = 7 - 3x^2$
• Second derivative: $\frac{d^2y}{dx^2} = -6x$
• Given rate of change of $x$: $\frac{dx}{dt} = 2 \text{ units/sec}$

Substituting the values at $x = 5$:

$$\frac{dm}{dt} = (-6 \cdot 5) \cdot 2$$

$$\frac{dm}{dt} = -30 \cdot 2 = -60 \text{ units/sec}$$

Step 3 (Final Answer): [1/4 Mark]

The rate at which the slope of the curve is changing is $-60$ units/sec.

Correct Option: (A)