Final Answer:
The final answer, 3sec²(3t), results from applying the chain rule to differentiate tan(3t), yielding sec²(3t) and multiplying by the derivative of the inner function, which is 3.Therefore the correct option is b.
Step-by-step explanation:
The derivative of y = tan(3t) involves applying the chain rule. To find dy/dt, start by differentiating the outer function, tan(3t), and then multiply by the derivative of the inner function, which is 3t. The derivative of tan(u) is sec²(u), so when the chain rule is applied, dy/dt = sec²(3t) d(3t)/dt. The derivative of 3t with respect to t is simply 3. Therefore, dy/dt = 3 sec²(3t).
When differentiating a trigonometric function like tan(3t), the chain rule becomes crucial. The derivative of tan(u) is sec²(u). Hence, for y = tan(3t), the derivative dy/dt requires applying the derivative of tan(3t) first, resulting in sec²(3t). Then, considering the derivative of the inner function (3t), which is 3, this value is multiplied to the previously obtained derivative of the outer function. Consequently, dy/dt = 3 sec²(3t), indicating that the rate of change of y with respect to t is three times sec²(3t).
In essence, the differentiation of y = tan(3t) incorporates the chain rule, wherein the derivative of the outer function (tan) yields sec²(3t), and the derivative of the inner function (3t) with respect to t results in 3. These derivatives, when combined using the chain rule, produce the final derivative dy/dt = 3 * sec²(3t), demonstrating the rate of change of y concerning t.Thus the correct option is b.