Find $$ \frac{d w}{d t} $$ using the appropriate Chain Rule for $$ w=x^{2}+y^{2} $$ where $$ x=7 t $$ and $$ y=3 t $$

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**Step 1. Express the Problem Using the Chain Rule**

We are given
$$
w = x^2 + y^2,
$$
with
$$
x = 7t \quad \text{and} \quad y = 3t.
$$
Using the chain rule, we have
$$
\frac{dw}{dt} = \frac{\partial w}{\partial x}\frac{dx}{dt} + \frac{\partial w}{\partial y}\frac{dy}{dt}.
$$

**Step 2. Compute the Partial Derivatives**

Differentiate $$w$$ with respect to $$x$$ and $$y$$:
$$
\frac{\partial w}{\partial x} = 2x,
$$
$$
\frac{\partial w}{\partial y} = 2y.
$$

**Step 3. Compute the Derivatives of $$x$$ and $$y$$ with Respect to $$t$$**

Differentiate $$x$$ and $$y$$ with respect to $$t$$:
$$
\frac{dx}{dt} = 7,
$$
$$
\frac{dy}{dt} = 3.
$$

**Step 4. Substitute All the Derivatives into the Chain Rule**

Substitute the computed derivatives into the chain rule formula:
$$
\frac{dw}{dt} = (2x)(7) + (2y)(3).
$$

**Step 5. Substitute $$x$$ and $$y$$ in Terms of $$t$$**

Since we know that $$x = 7t$$ and $$y = 3t$$, substitute these into the expression:
$$
\frac{dw}{dt} = 2(7t)(7) + 2(3t)(3).
$$

**Step 6. Simplify the Expression**

Simplify each term separately:
$$
2(7t)(7) = 98t,
$$
$$
2(3t)(3) = 18t.
$$
Thus,
$$
\frac{dw}{dt} = 98t + 18t = 116t.
$$

**Final Answer**

$$
\frac{dw}{dt} = 116t.
$$
$$ \frac{dw}{dt} = 116t $$

Find using the appropriate Chain Rule for where and

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