Question

Evaluate ddtr(g(t)) using the Chain Rule:
r(t)=〈et,e3t,−9〉,g(t)=8t−1
d/dtr(g(t))

Answer 1

d/dt r(g(t)) = <8e^(8t-1), 24e^(24t-3), 0>

Explanation:

To use the Chain Rule to evaluate d/dt r(g(t)), we need to first substitute g(t) into the components of the vector r(t) and then differentiate each component with respect to t.

So, we have:

r(g(t)) = r(8t - 1) = <e^(8t-1), e^(24t-3), -9>

Now, to find d/dt r(g(t)), we differentiate each component with respect to t:

d/dt r(g(t)) = <d/dt (e^(8t-1)), d/dt (e^(24t-3)), d/dt (-9)>

= <8e^(8t-1), 24e^(24t-3), 0>

Therefore, d/dt r(g(t)) = <8e^(8t-1), 24e^(24t-3), 0>.