Answer:
First Part:
exponential function [tex]e^t[/tex] is the one whose first order derivative is the function itself.
Second Part:
[tex]y=ce^{At}\\y'=Ay[/tex]
Third Part:
[tex]y=ce^t\\y'=ce^t\\y'=y[/tex]
Step-by-step explanation:
First Part:
In calculus exponential function [tex]e^t[/tex] is the one whose first order derivative is the function itself.
Where:
t is independent variable.
Derivative is represented as:
[tex]y=ce^t\\y'=\frac{d(ce^t)}{dt} \\y'=ce^t\\y'=y[/tex]
Where:
c is any number.
Second Part:
Consider the constant A.
The function will become:
[tex]y=ce^{At}\\y'=\frac{d(ce^{At})}{dt} \\y'=cAe^{At}\\y'=Ay[/tex]
Third Part:
Derivative is represented as:
[tex]y=ce^t\\y'=\frac{d(ce^t)}{dt} \\y'=ce^t\\y'=y[/tex]
Where:
c is any number.
