Question

question if (y = - 5x^{2}-15) and (\frac{dx}{dt}=3), find (\frac{dy}{dt}) at (x = - 1).

Explanation:

Step1: Differentiate y with respect to x

Given $y = - 5x^{2}-15$, using the power - rule $\frac{d}{dx}(ax^{n})=nax^{n - 1}$, we have $\frac{dy}{dx}=-10x$.

Step2: Evaluate $\frac{dy}{dx}$ at $x = - 1$

Substitute $x=-1$ into $\frac{dy}{dx}$, so $\frac{dy}{dx}\big|_{x = - 1}=-10\times(-1) = 10$.

Step3: Use the chain - rule $\frac{dy}{dt}=\frac{dy}{dx}\cdot\frac{dx}{dt}$

We know that $\frac{dx}{dt}=3$. Then $\frac{dy}{dt}=\frac{dy}{dx}\cdot\frac{dx}{dt}$.
Substitute $\frac{dy}{dx}\big|_{x = - 1}=10$ and $\frac{dx}{dt}=3$ into the chain - rule formula, we get $\frac{dy}{dt}=10\times3 = 30$.

Step4: Find $\frac{dt}{dy}$

Since $\frac{dt}{dy}=\frac{1}{\frac{dy}{dt}}$, and $\frac{dy}{dt}=30$, then $\frac{dt}{dy}=\frac{1}{30}$.

Answer:

$\frac{1}{30}$