question: if $y = x^{2}-6$ and $\frac{dx}{dt}=3$, find $\frac{dy}{dt}$ at $x = - 5$. provide your answer…

question: if $y = x^{2}-6$ and $\frac{dx}{dt}=3$, find $\frac{dy}{dt}$ at $x = - 5$. provide your answer below: $\frac{dy}{dt}=square$ 4.1 homework first up, lets review the assignments learning objectives. get familiar with this topic by reviewing instruction and answering a couple of questions.

question: if $y = x^{2}-6$ and $\frac{dx}{dt}=3$, find $\frac{dy}{dt}$ at $x = - 5$. provide your answer below: $\frac{dy}{dt}=square$ 4.1 homework first up, lets review the assignments learning objectives. get familiar with this topic by reviewing instruction and answering a couple of questions.

Answer

Explanation:

Step1: Differentiate y with respect to x

Using the power - rule, if $y = x^{2}-6$, then $\frac{dy}{dx}=2x$.

Step2: Use the chain - rule

The chain - rule states that $\frac{dt}{dy}=\frac{dt}{dx}\cdot\frac{dx}{dy}$. We know that $\frac{dt}{dx} = 3$ and $\frac{dx}{dy}=\frac{1}{\frac{dy}{dx}}$.

Step3: Substitute $\frac{dy}{dx}$ into the chain - rule formula

Since $\frac{dy}{dx}=2x$, then $\frac{dx}{dy}=\frac{1}{2x}$. And $\frac{dt}{dy}=\frac{dt}{dx}\cdot\frac{dx}{dy}=3\cdot\frac{1}{2x}=\frac{3}{2x}$.

Step4: Evaluate at x = - 5

Substitute $x=-5$ into $\frac{dt}{dy}$. We get $\frac{dt}{dy}\big|_{x = - 5}=\frac{3}{2\times(-5)}=-\frac{3}{10}$.

Answer:

$-\frac{3}{10}$