Asksia AI LOGO

Sia

Question
Math
Posted 6 months ago

Suppose we have a vector-valued function g(t)g(t) and a scalar function f(x,y)f(x, y). Let h(t)=f(g(t))h(t)=f(g(t)).

We know:
g(2)=(4,2)g(2)=(1,3)f(4,2)=(6,2)
\begin{array}{l}
g(2)=(-4,2) \\
g^{\prime}(2)=(1,3) \\
\nabla f(-4,2)=(6,-2)
\end{array}


Evaluate dhdt\frac{d h}{d t} at t=2t=2.
h(2)=
h^{\prime}(2)=
Sign in to unlock the answer
Answer from Sia
Posted 6 months ago
Solution by Steps
step 1
To find the derivative of the composite function h(t)=f(g(t))h(t) = f(g(t)), we use the chain rule
step 2
The chain rule states that h(t)=f(g(t))g(t)h'(t) = \nabla f(g(t)) \cdot g'(t)
step 3
We substitute the given values into the chain rule formula: f(4,2)=(6,2)\nabla f(-4,2) = (6,-2) and g(2)=(1,3)g'(2) = (1,3)
step 4
We calculate the dot product of f(4,2)\nabla f(-4,2) and g(2)g'(2): (6,2)(1,3)=61+(2)3(6,-2) \cdot (1,3) = 6*1 + (-2)*3
step 5
Simplifying the dot product: 61+(2)3=66=06*1 + (-2)*3 = 6 - 6 = 0
Answer
h(2)=0h'(2) = 0
Key Concept
Chain Rule for Derivatives
Explanation
The derivative of a composite function h(t)=f(g(t))h(t) = f(g(t)) at a point is found by taking the dot product of the gradient of ff at g(t)g(t) and the derivative of gg at tt.

Not the question you are looking for? Ask here!

Enter question by text

Enter question by image

Unlock Smarter Learning with AskSia Super!

Join Super, our all-in-one AI solution that can greatly improve your learning efficiency.

30% higher accuracy than GPT-4o
Entire learning journey support
The most student-friendly features
Study Other Question