Approximate Calculations Using Differentials
Big Idea
When exact computation is difficult, calculus gives us a powerful shortcut: we can use the tangent line at a known point to approximate function values at nearby points. This technique, called linear approximation, is one of the most practical applications of derivatives.
1. The Differential
For a differentiable function $y = f(x)$, the differential $dy$ is defined as:
$$ dy = f'(x) \cdot dx $$
where $dx = \Delta x$ is a small change in $x$. The key insight is:
- $\Delta y = f(x + \Delta x) - f(x)$ is the actual change in $y$
- $dy = f'(x) \cdot dx$ is the approximate change in $y$ (along the tangent line)
When $\Delta x$ is small: $\Delta y \approx dy$, which means:
$$ f(x + \Delta x) \approx f(x) + f'(x) \cdot \Delta x $$
This is the Linear Approximation Formula (also called the tangent line approximation).