Question

question: if y = 2x^2 + 13 and dx/dt=-2, find dy/dt at x = - 5. provide your answer below: dy/dt =

Explanation:

Step1: Differentiate y with respect to x

Using the power - rule, if $y = 2x^{2}+13$, then $\frac{dy}{dx}=4x$.

Step2: Use the chain - rule

The chain - rule states that $\frac{dy}{dt}=\frac{dy}{dx}\cdot\frac{dx}{dt}$. We want to find $\frac{dy}{dt}$, and we know that $\frac{dx}{dt}=-2$. So $\frac{dy}{dt}=(4x)\cdot(-2)=-8x$.

Step3: Evaluate at x = - 5

Substitute $x = - 5$ into the equation for $\frac{dy}{dt}$. When $x=-5$, $\frac{dy}{dt}=-8\times(-5)=40$.

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

Since $\frac{dy}{dt} = 40$, then $\frac{dt}{dy}=\frac{1}{\frac{dy}{dt}}=\frac{1}{40}$.

Answer:

$\frac{1}{40}$