Differentiability at a point x=0

Question

I road in a book that the function $f(x)=\sqrt{x} \sin(x)$ is differentiable at 0. Personally, I don't think so since f is undefined on the left of $x=0$. Can you confirm my thought or the book?

Answer 1

Since your domain is $[0,\infty )$ we only have one sided derivative and that is what the book meant by differentiability at $0.$

Now to see if f(x) is indeed differentiable, we look at the difference quotient $$\frac{f(x)-f(0)}{x-0}$$ and let $x\to 0^+$. Since $f(0)=0$ the difference quotient simplifies to $$ \frac{sin(x)}{x} \sqrt {x}$$ which approaches $1\times 0=0$ as $x\to 0^+$.