Question

If $S = 12t - 3t^2$ then $\frac{dS}{dt} =$

• A. $12 - 6t$
• B. $12t - 6$
• C. $12 - 3t$
• D. $12 - 6t^2$

Hint:

Always keep track of the powers. A linear term ($t$) becomes a constant, and a quadratic term ($t^2$) becomes a linear term. This "power reduction" is the essence of the power rule.

Answer 1

The Correct Option is A

1. The Function: The given expression for displacement (or position) $S$ is a quadratic function of time $t$: $$S(t) = 12t - 3t^2$$

2. Differentiating Term by Term: Apply the rule $\frac{d}{dt}(ct^n) = cn \cdot t^{n-1}$:

• Derivative of $12t$: Since the power of $t$ is 1, $\frac{d}{dt}(12t^1) = 12(1)t^0 = 12$.

• Derivative of $-3t^2$: Using the power rule, $\frac{d}{dt}(-3t^2) = -3(2)t^{2-1} = -6t$.

3. Final Result: Combining the results: $$\frac{dS}{dt} = 12 - 6t$$ In physical terms, if $S$ represents displacement, then $\frac{dS}{dt}$ represents the instantaneous velocity at time $t$.