Derivative of e^y: Solving w/ Chain Rule

[brambleberry]

Homework Statement

What is the derivative of e^y? i think i am differentiating with respect to x

Homework Equations

Derivative of y^x is y^x

The Attempt at a Solution

I don't know if I should use the chain rule or treat it like y^x. When i used the chain rule I got ye^y-1, but then I wondered if it should be e^y.

 

[Dick]

The derivative with respect to y? Then sure, d/dy(e^y)=e^y. You can't use the power law d/dy(y^n)=n*y^(n-1). In one the variable y is an exponent, in the other it's not. They are very different functions.

 

[HallsofIvy]

Homework Statement

What is the derivative of e^y? i think i am differentiating with respect to x

Homework Equations

Derivative of y^x is y^x

The Attempt at a Solution

I don't know if I should use the chain rule or treat it like y^x. When i used the chain rule I got ye^y-1, but then I wondered if it should be e^y.

I don't see what y^x has to do with your original equation. y^x is not anything like e^y and yes, you should use the chain rule. But the chain rule does NOT give "ye^{y-1}"!

The chain rule says that
$$\frac{d e^y}{dx}= \frac{de^y}{dy}{dy}{dx}$$
$$\frac{d e^y}{dy}$$
is $e^y$, NOT "$ye^{y-1}$". That power formula only applies to the variable to a constant power, not a constant power to a variable power.

[tex]\frac{de^y}{dx}= \frac{de^y}{dy}\frac{dy}{dx}= e^y\frac{dy}{dx}[/itex][/tex]