Question

Math

Posted 3 months ago

```
Suppose we have a vector-valued function $g(t)$ and a scalar function $f(x, y)$. Let $h(t)=f(g(t))$.
We know:
$\begin{array}{l}
g(2)=(-4,2) \\
g^{\prime}(2)=(1,3) \\
\nabla f(-4,2)=(6,-2)
\end{array}$
Evaluate $\frac{d h}{d t}$ at $t=2$.
$h^{\prime}(2)=$
```

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Answer from Sia

Posted 3 months ago

Solution by Steps

step 1

To find the derivative of the composite function $h(t) = f(g(t))$, we use the chain rule

step 2

The chain rule states that $h'(t) = \nabla f(g(t)) \cdot g'(t)$

step 3

We substitute the given values into the chain rule formula: $\nabla f(-4,2) = (6,-2)$ and $g'(2) = (1,3)$

step 4

We calculate the dot product of $\nabla f(-4,2)$ and $g'(2)$: $(6,-2) \cdot (1,3) = 6*1 + (-2)*3$

step 5

Simplifying the dot product: $6*1 + (-2)*3 = 6 - 6 = 0$

Answer

$h'(2) = 0$

Key Concept

Chain Rule for Derivatives

Explanation

The derivative of a composite function $h(t) = f(g(t))$ at a point is found by taking the dot product of the gradient of $f$ at $g(t)$ and the derivative of $g$ at $t$.

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