Question

What function do you know from calculus is such that its first derivative is itself? (Do not use the function f(x) = 0.)

Answer 1

Answer:

$f(x)=e^x$

Step-by-step explanation:

Proof:

Let $e^x=z$

take logs in both sides$ln(e^x)=ln(z)$

using log properties we have $xln(e)=ln(z)$. We know that $ln(e)=1$ then $x=ln(z)$

Taking derivatives with respect to x, we have: $1=\frac{1}{z}\frac{dz}{dx}$

Next we can move things around $z=\frac{dz}{dx}$

Replacing z for $e^x$ as defined in the first line, we get

$e^x=\frac{d}{dx}e^x$