This parametric curve requires applying $ \frac{dy}{dx} = \frac{dy/dt}{dx/dt} $ to find the derivative. For $ x(t) = e^t $, we have $ dx/dt = e^t $. For $ y(t) = t e^t $, we use the product rule: $ dy/dt = e^t + t e^t $. The chain rule then gives $ dy/dx = (e^t + t e^t) / e^t $. Choice B incorrectly inverts this ratio by placing $ dx/dt $ in the numerator instead of the denominator. The essential parametric differentiation approach is to compute $ dy/dt $ and $ dx/dt $ separately, then divide the first by the second.

This parametric curve applies dy/dx = (dy/dt)/(dx/dt). For x(t) = t² + t/2, we calculate dx/dt = 2t + 1/2. For y(t) = t² - t/2, we get dy/dt = 2t - 1/2. Using the chain rule yields dy/dx = (2t - 1/2)/(2t + 1/2). Choice B incorrectly inverts the fraction by putting dx/dt in the numerator. The essential parametric differentiation approach is to compute both component derivatives, then divide dy/dt by dx/dt.

This parametric curve requires $dy/dx = (dy/dt)/(dx/dt)$. For $x(t) = \frac{1}{2}t^2$, we find $dx/dt = t$. For $y(t) = \frac{1}{4}t^4$, we get $dy/dt = t^3$. Using the parametric derivative formula yields $dy/dx = t^3/t = t^2$. Choice A shows this result correctly. The standard parametric differentiation method involves finding derivatives of both components, then dividing $dy/dt$ by $dx/dt$.

This parametric curve uses $\frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}}$. For $x(t) = t^2 + \tan t$, we calculate $\frac{dx}{dt} = 2t + \sec^2 t$. For $y(t) = t^2 + \sec t$, we get $\frac{dy}{dt} = 2t + \sec t \tan t$. Using the parametric derivative gives $\frac{dy}{dx} = \frac{2t + \sec t \tan t}{2t + \sec^2 t}$. Choice B incorrectly inverts the fraction. The fundamental parametric differentiation technique is to compute derivatives of both components, then divide $\frac{dy}{dt}$ by $\frac{dx}{dt}$.

This parametric differentiation uses the chain rule dy/dx = (dy/dt)/(dx/dt). For x(t) = sin(2t), we find dx/dt = 2cos(2t). For y(t) = cos(3t), we get dy/dt = -3sin(3t). The parametric derivative formula gives dy/dx = (-3sin(3t))/(2cos(2t)). Choice B incorrectly inverts the fraction by placing dx/dt in the numerator. The key parametric differentiation strategy is to compute both component derivatives, then form the ratio dy/dt over dx/dt.

This problem involves parametric differentiation, where we find the derivative dy/dx for curves defined by parametric equations x(t) and y(t). To compute dy/dx using the chain rule, recognize that dy/dx = (dy/dt) / (dx/dt), provided dx/dt ≠ 0. First, differentiate x(t) = sec t to get dx/dt = sec t tan t, and y(t) = tan t to get dy/dt = sec² t. Then, divide these derivatives to obtain dy/dx = sec² t / (sec t tan t). A tempting distractor might be sec t tan t / sec² t, which fails because it incorrectly inverts the ratio of the derivatives. Always remember the general strategy for parametric derivatives: compute dy/dt and dx/dt separately, then form their quotient for dy/dx.

This problem involves parametric differentiation, where we find the derivative dy/dx for curves defined by parametric equations x(t) and y(t). To compute dy/dx using the chain rule, recognize that dy/dx = (dy/dt) / (dx/dt), provided dx/dt ≠ 0. First, differentiate x(t) = t² - 3t to get dx/dt = 2t - 3, and y(t) = 4t + sin t to get dy/dt = 4 + cos t. Then, divide these derivatives to obtain dy/dx = (4 + cos t) / (2t - 3). A tempting distractor might be (2t - 3) / (4 + cos t), which fails because it incorrectly inverts the ratio of the derivatives. Always remember the general strategy for parametric derivatives: compute dy/dt and dx/dt separately, then form their quotient for dy/dx.

This parametric differentiation uses dy/dx = (dy/dt)/(dx/dt). For x(t) = t² + t, we calculate dx/dt = 2t + 1. For y(t) = t²/2 + t, we get dy/dt = t + 1. Using the chain rule gives dy/dx = (t + 1)/(2t + 1). Choice B incorrectly inverts the fraction by putting dx/dt in the numerator. The key parametric derivative technique is to differentiate each component separately, then form the ratio with dy/dt on top.

This parametric differentiation problem uses dy/dx = (dy/dt)/(dx/dt). For x(t) = t² - 4, we calculate dx/dt = 2t. For y(t) = 1/t, we get dy/dt = -1/t². The chain rule gives dy/dx = (-1/t²)/(2t) = -1/(2t³). Choice B incorrectly inverts the fraction by putting dx/dt in the numerator. The fundamental parametric derivative approach is to differentiate each component separately, then divide dy/dt by dx/dt.

This parametric differentiation requires dy/dx = (dy/dt)/(dx/dt). For x(t) = t³, we calculate dx/dt = 3t². For y(t) = ln(t), we get dy/dt = 1/t. The chain rule gives dy/dx = (1/t)/(3t²) = 1/(3t³). Choice B incorrectly inverts the fraction by putting dx/dt in the numerator. The reliable parametric derivative strategy is to find derivatives of both components, then form their proper quotient.