Question

question: if (y = x^{4}-4) and (\frac{dx}{dt}=-8), find (\frac{dy}{dt}) at (x = - 4). provide your answer below: (\frac{dy}{dt}=)

Explanation:

Step1: Differentiate y with respect to x

Using the power - rule, if $y = x^{4}-4$, then $\frac{dy}{dx}=4x^{3}$.

Step2: Use the chain - rule

The chain - rule states that $\frac{dy}{dt}=\frac{dy}{dx}\cdot\frac{dx}{dt}$, so $\frac{dt}{dy}=\frac{1}{\frac{dy}{dt}}=\frac{1}{\frac{dy}{dx}\cdot\frac{dx}{dt}}$.

Step3: Substitute x = - 4 into $\frac{dy}{dx}$

When $x=-4$, $\frac{dy}{dx}=4(-4)^{3}=4\times(-64)=-256$.

Step4: Substitute $\frac{dy}{dx}$ and $\frac{dx}{dt}$ into $\frac{dt}{dy}$

Given $\frac{dx}{dt}=-8$, then $\frac{dt}{dy}=\frac{1}{\frac{dy}{dx}\cdot\frac{dx}{dt}}=\frac{1}{(-256)\times(-8)}=\frac{1}{2048}$.

Answer:

$\frac{1}{2048}$