Functions: $f\left(u,v\right)=u^{2}+3v^{2}$
$c\left(t\right)=\begin{pmatrix} e^{t} \\ e^{-t} \end{pmatrix} $
I calculate composition and drivative on 2 ways: 1. substitution and 2. chain rule. But I always got different results and just cant find my mistake. Here are my steps.
- Substitution: $ f\circ c=f\begin{pmatrix} e^{t} \\ e^{-t} \end{pmatrix} =4\left(e^{t}\right)^{2} $
$\frac{df}{dt}=8t\left(e^{t}\right)^{2} $
- Chain rule
$\frac{df}{dt}=\frac{\partial f}{\partial u} \frac{du}{dt}+\frac{\partial f}{\partial v} \frac{dv}{dt}=2e^{t}e^{t}-6e^{-t}e^{-t}=2e^{2t}-6e^{-2t}$
I would really appreciate if someone can say me where I am making mistake here. Why I am not getting the same result with both ways.
$f\begin{pmatrix} e^{t} \\ e^{-t} \end{pmatrix}=(e^t)^2+3(e^{-t})^2=e^{2t}+3e^{-2t}$. Now take the derivative. Your mistake was that you took $(e^{-t})^2=e^{2t}$, which is not true. In general $(e^m)^n=e^{mn.}$