Question

question: if (y = x^{2}+5) and (\frac{dx}{dt}=-2), find (\frac{dy}{dt}) at (x = 0). provide your answer below: (\frac{dy}{dt}=square)

Explanation:

Step1: Differentiate y with respect to x

Using the power - rule, if $y = x^{2}+5$, then $\frac{dy}{dx}=2x$.

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 $\frac{dx}{dt}=-2$. Substituting $\frac{dy}{dx}=2x$ and $\frac{dx}{dt}=-2$ into the chain - rule formula, we get $\frac{dy}{dt}=(2x)\cdot(-2)=-4x$.

Step3: Evaluate at x = 0

Substitute $x = 0$ into the equation for $\frac{dy}{dt}$. When $x = 0$, $\frac{dy}{dt}=-4\times0 = 0$.

Answer:

0