Let's solve this problem step-by-step.

Given: $f(x) = \sqrt[3]{x}$ we are asked to use the definition of the derivative: $f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}$ to find $f'(a)$ when $a \neq 0$, and then to show that $f'(0)$ does not exist, indicating a vertical tangent line at $(0,0)$.

Step 1: Finding $f'(a)$ for $a \neq 0$

1. Substitute $f(x) = \sqrt[3]{x}$ into the derivative formula: [ f(a) = \sqrt[3]{a} \quad \text{and} \quad f(a + h) = \sqrt[3]{a + h} ] so [ f'(a) = \lim_{h \to 0} \frac{\sqrt[3]{a + h} - \sqrt[3]{a}}{h}. ]

2. Simplify the Expression Using a Conjugate Technique: To handle the cube root, let’s multiply the numerator and denominator by the conjugate factor: [ f'(a) = \lim_{h \to 0} \frac{\sqrt[3]{a+h} - \sqrt[3]{a}}{h}. ]