Final answer:
To calculate (f∘c)'(t), where f(x,y) = 9e^(xy) and c(t) = (6t^2, t^7), use the first special case of the chain rule for composition to find the derivative.
Step-by-step explanation:
To calculate the derivative of (f∘c)(t), where f(x,y) = 9e^(xy) and c(t) = (6t^2, t^7), we need to use the first special case of the chain rule for composition. The chain rule states that the derivative of (f∘c)(t) is equal to the derivative of f with respect to x times the derivative of c with respect to t, plus the derivative of f with respect to y times the derivative of c with respect to t.
First, let's find the derivative of f with respect to x. Since f(x,y) = 9e^(xy), the derivative of f with respect to x is equal to 9ye^(xy).
Next, let's find the derivative of c with respect to t. Since c(t) = (6t^2, t^7), the derivative of c with respect to t is equal to (12t, 7t^6).
Now we can calculate (f∘c)'(t) using the chain rule:
(f∘c)'(t) = (9ye^(xy) * 12t) + (9e^(xy) * 7t^6)