Mathematics

Mathematics

Problem

Let kk be a non-zero constant. The curve is defined by x=t2+2ktx = t^2 + 2k t and y=kt3−1y = k t^3 - 1. Determine dydx\dfrac{dy}{dx} at t=−kt = -k in terms of kk.
Intuitive explanation

You're finding the derivative of a parametric curve, which means you need dydx=dy/dtdx/dt\frac{dy}{dx} = \frac{dy/dt}{dx/dt} rather than differentiating yy with respect to xx directly. The key is to compute both derivatives separately, then evaluate them at the specific parameter value t=−kt = -k—this substitution often simplifies expressions that might look complicated at first.

Intuitive explanation

You're finding the derivative of a parametric curve, which means you need dydx=dy/dtdx/dt\frac{dy}{dx} = \frac{dy/dt}{dx/dt} rather than differentiating yy with respect to xx directly. The key is to compute both derivatives separately, then evaluate them at the specific parameter value t=−kt = -k—this substitution often simplifies expressions that might look complicated at first.

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