Finding values of a piecewise function such that it is differentiable at $x=1$

Question

Let

$$ f(x)= \begin{cases} a-x & x \leq 1, \\ \frac{1}{bx} & x>1. \end{cases} $$

Considering this piecewise defined function find values of $a,b$ such that the function is differentiable at $x=1$. Give the value of $f'(1)$.

I don't know how to go about this question without having any actual numerical values. Any help would be greatly appreciated.

Answer 1

Certainly it has to be continuous at $x=1$, meaning that $a-x = 1/(bx)$ for $x=1$, or $a-1 = 1/b$.

Also, the derivative from the left has to match the derivative from the right, meaning that

$$-1 = -\frac{1}{bx^2}$$

at $x=1$, or

$$-1 = -\frac{1}{b}.$$