The differentiable functions \[x\] and \[y\] are related by the following equation:

\[y=2x^2-x\]

Also, \[\dfrac{dx}{dt}=-0.1\].

**Find \[\dfrac{dy}{dt}\] when \[x=-3\].**

To find \(\dfrac{dy}{dt}\) when \(x = -3\), we need to use the chain rule, given that \(y\) is a function of \(x\), and \(x\) is a function of \(t\).

### Step 1: Differentiate \(y = 2x^2 - x\) with respect to \(t\)

Using the chain rule, the derivative of \(y\) with respect to \(t\) is:
\[
\frac{dy}{dt} = \frac{dy}{dx} \cdot \frac{dx}{dt}
\]

### Step 2: Find \(\frac{dy}{dx}\)

First, we need to differentiate \(y = 2x^2 - x\) with respect to \(x\):
\[
\frac{dy}{dx} = \frac{d}{dx}(2x^2 - x) = 4x - 1
\]

### Step 3: Substitute \(x = -3\) and \(\frac{dx}{dt} = -0.1\)

Now, substitute \(x = -3\) into \(\frac{dy}{dx}\):
\[
\frac{dy}{dx} \bigg|_{x=-3} = 4(-3) - 1 = -12 - 1 = -13
\]

### Step 4: Compute \(\frac{dy}{dt}\)

Now, substitute \(\frac{dy}{dx} = -13\) and \(\frac{dx}{dt} = -0.1\) into the chain rule formula:
\[
\frac{dy}{dt} = (-13) \cdot (-0.1) = 1.3
\]

### Final Answer

Therefore, \(\dfrac{dy}{dt}\) when \(x = -3\) is:
\[
\frac{dy}{dt} = 1.3
\]

Would you like to explore any of the steps in more detail? Here are some related questions you might be interested in:

1. How does the chain rule work when multiple variables are involved?
2. What would \(\dfrac{dy}{dt}\) be if \(\dfrac{dx}{dt}\) were a different value?
3. How does the sign of \(\dfrac{dx}{dt}\) affect the overall result?
4. Can you find \(\dfrac{d^2y}{dt^2}\) for this function?
5. How would you interpret the physical meaning of \(\dfrac{dy}{dt}\) in a real-world scenario?

**Tip:** When working with related rates, always keep track of which variables are dependent on others and apply the chain rule carefully.

Learn how to find dy/dt when x = -3 using differentiation and the chain rule. Step-by-step solution provided for the function y = 2x^2 - x with dx/dt = -0.1. Suitable for advanced high school students and learners interested in calculus.

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