Question

f(x,y)=x^2+y and c(t)=e^t,e^-t
write the composition f(c(t)) as a function of t, then differentiating with respect to t

Answer 1

When c(t) = e^t,
f(c(t)) = e^(2t)
Differentiating e^(2t) w.r.t. t,
2e^(2t)
When c(t) = e^-t,
f(c(t)) = e^(-2t)
Differentiating e^(-2t) w.r.t. t,
-2e^(-2t)

Answer 2

ok i see! Then to compute d/dt f(c(t)) i use the what I differentiated w.r.t. t?

Answer 3

I have already differentiated with respect to t. f(c(t)) just means that where x is in f(x) you just replace it with the value of c(t). So by doing this you get a function dependent on t, which like you said, you differentiate w.r.t. to t.

Answer 4

oh now i see what you mean. Thank you!

Answer 5

Oh i'm really really sorry.
f(c(t)) means where there is x substitute e^t and where there is y substitute e*-t

Answer 6

e^-t for y.

Answer 7

oh ok! so I replace the x and the y with the c(t) values and differentiate w.r.t. t?

Answer 8

so then f(c(t)) becomes
e^2t+e(-t)
Now differentiate it with respect to t,
we get
2e^2t-e^-t

Answer 9

ok that makes more sense! so if i wanted to compute d/dt f(c(t)) using the chain rule would i use the formula d/dt f(c(t))=df/dx(dx/dt)+df/dy(dy/dt)

Answer 10

same f(x,y) and C(t)

Answer 11

yes, exactly. Either way works! :)