Help with dummy variables an proving properties of the exponential function.

Question

I am considering a function, $\exp(x)$ such that

• $\frac{d}{dx}\exp(x)=\exp(x)$
• $\exp(0)=1$

From these two I am trying to prove that such a function would have properties reminiscent of powers, specifically that $\exp(u+v)=\exp(u)\exp(v)$

The derivation I am looking at is here:

However I am very confused as to what the step is where $w'$ has been introduced. I don't understand why $exp(u)$ suddenly appears in the limits of the second integral. Any help woould be much appreciated. I have tried looking up dummy variables online and various searches about the exponential function, but I couldn't find anything about this.

Answer 1

That's because $$w=1 \Leftrightarrow w'=\exp(u)$$ $$ w=\exp(v) \Leftrightarrow w'=\exp(u) \cdot \exp(v)$$ You just need to change the integration bounds after making the substitution.

Answer 2

It's just a standard substitution. Notice that we are looking at

$$\int_1^{\exp(v)}\frac{dw}w=\int_{\exp(u)}^{\exp(v)\exp(u)}\frac{dw'}{w'}$$

For the bounds, notice that the lower bound says $w=1$. On the next integral, the lower bound is $w'=w\exp(u)=\exp(u)$, since $w=1$. Same thing happens for the upper bound of the integral.