Respuesta :
Answer:
[tex] \frac{d}{dx} (e^{\frac{1}{x}}) = e^{\frac{1}{x}} (-\frac{1}{x^2}) = -\frac{e^{\frac{1}{x}}}{x^2}[/tex]
Step-by-step explanation:
Assuming the following function [tex] y = e^{\frac{1}{x}}[/tex] we want to find the derivate of this function.
For this case we need to apply the chain rule given by the following formula:
[tex] \frac{df(u)}{dx} = \frac{df}{du} \frac{du}{dx}[/tex]
On this case our function is [tex] f = e^u[/tex] and our value for u is [tex] u = \frac{1}{x}[/tex]
If we appply this rule we got this:
[tex]\frac{df(u)}{dx} = \frac{d}{du} (e^u) \frac{d}{dx} (\frac{1}{x})[/tex]
[tex] \frac{df(u)}{dx} = e^u (-\frac{1}{x^2})[/tex]
And now w can substitute [tex] u = \frac{1}{x}[/tex] and we got:
[tex] \frac{d}{dx} (e^{\frac{1}{x}}) = e^{\frac{1}{x}} (-\frac{1}{x^2}) = -\frac{e^{\frac{1}{x}}}{x^2}[/tex]
