# Tag Info

33

Assume that $f(x)$ is a function such that $f'(x)=f(x)$ for all $x\in\Bbb{R}$. Consider the quotient $g(x)=f(x)/e^x$. We can differentiate $$g'(x)=\frac{f'(x)e^x-f(x)D e^x}{(e^x)^2}=\frac{f(x)e^x-f(x)e^x}{(e^x)^2}=0.$$ By the mean value theorem it follows that $g(x)$ is a constant. QED.

10

Like square root, log is treated as a "multi-valued function." It's just that when we restrict to the positive real line, we can make it single-valued. (Just as we do with square root in algebra 1.) So the first few things you derived are right: one of the logarithms of $-1$ is $\pi i$. The others all differ from it by multiples of $2\pi i$. So when you ...

8

Consider the equation $y'=y$. Our goal is to solve for the function $y=f(x)$. Roughly speaking $$\frac{dy}{dx}=y \implies \frac{dy}{y}=dx \implies \int\frac{dy}{y}=\int dx \implies\ln(y)=x+C \implies y=e^{x+C}=Ae^x$$ for some constant $A$

5


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