What is the rigorous mathematical definition of the derivative at a point x₀ expressed as the limit of the difference quotient?

Conditions

• The limit exists and is finite.
• $f(x)$ is defined in a neighborhood of $x_0$.

Reasoning, step by step

1. Start with the average rate of change formula over an interval $\Delta x$: $\frac{f(x_0+\Delta x)-f(x_0)}{\Delta x}$.
2. Apply the limit operator as $\Delta x \to 0$.
3. Equate this limit to the notation $f'(x_0)$.
4. Interpret the result as the slope of the tangent line at the specific point $x_0$.

Example

The video displays the final formula in the top-left corner: $\lim_{\Delta x\to0} [f(x_0+\Delta x)−f(x_0)]/\Delta x = f’(x_0)$, annotated as ‘slope of tangent.’

Common misconceptions

• Confusing the derivative $f'(x_0)$ with the differential $dy$.
• Assuming the derivative is always equal to the function value $f(x_0)$.

Watch the explanation

Explore next

• How does this definition relate to the geometric concept of the tangent line?
• What happens if the limit does not exist?