To differentiate $\left(\frac{f}{g}\right)'$, we use the quotient rule for derivatives. The quotient rule states that for two differentiable functions $f(x)$ and $g(x)$, the derivative of their quotient is given by:

$\left(\frac{f}{g}\right)' = \frac{g(x)f'(x) - f(x)g'(x)}{(g(x))^2}$

Steps:

1. Differentiate the numerator function $f(x)$ to get $f'(x)$.
2. Differentiate the denominator function $g(x)$ to get $g'(x)$.
3. Apply the quotient rule formula.

Would you like to see this applied to specific functions $f(x)$ and $g(x)$? Let me know if you'd like more details or examples.

Related Questions:

1. What happens if $g(x) = 1$ in the quotient rule?
2. How does the quotient rule differ from the product rule in calculus?
3. Can you apply the quotient rule to trigonometric functions?
4. How would you use the quotient rule in implicit differentiation?
5. How does the quotient rule handle higher-order derivatives?

Tip:

When using the quotient rule, always make sure that $g(x) \neq 0$ because division by zero is undefined.