Deriving the Linearization Formula

From a Tangent Line to a Linearization

A differentiable function can often be represented near a chosen input by a line that follows the function at that input. The line is not generally equal to the function at nearby inputs, but it provides a simple local model. To construct it, we need the function’s value and slope at one specified input, called the base point.

The formula comes directly from two familiar facts about a tangent line: its slope at the base point is the derivative there, and it passes through the point on the graph at that input. Combining those facts gives a formula that can be applied systematically.

Deriving the Formula

Let \(f\) be differentiable at a point \(a\) in its domain. The graph of \(f\) contains the point \((a,f(a))\), and the tangent line to the graph there has slope \(f'(a)\). A line of slope \(m\) through \((x_0,y_0)\) has point-slope form \(y-y_0=m(x-x_0)\). Use \(m=f'(a)\), \(x_0=a\), and \(y_0=f(a)\). The tangent line is therefore

Solving for \(y\) gives the line’s output as a function of \(x\). We call this line the linearization of \(f\) at \(a\).

This expression is the tangent line written as a function of \(x\). The two terms have distinct roles: \(f(a)\) places the line at the correct height, while \(f'(a)(x-a)\) changes its output according to the tangent slope and the horizontal displacement from \(a\).

Why This Is a Local Model

The formula does more than describe a line with the right value and slope. Differentiability explains why that line models the function near the base point. By the definition of the derivative, for \(x\ne a\),

Subtract the tangent-line change \(f'(a)(x-a)\) from the actual change \(f(x)-f(a)\). The difference is

For \(x\ne a\), factor out \(x-a\) and divide the remaining factor by \(x-a\):

The expression in parentheses tends to zero as \(x\) approaches \(a\), by differentiability. Thus the difference between the function and its linearization is \(x-a\) multiplied by a quantity that tends to zero. In particular, the error is smaller than a fixed multiple of \(|x-a|\) sufficiently close to \(a\). This is the precise first-order sense in which the tangent line models the function locally.

A Reliable Construction Process

For a specified function and base point, the calculation has three essential ingredients: evaluate the function at the base point, evaluate the derivative there, and place both values in the formula. The following sequence keeps those ingredients visible.

Worked Examples

Conditions and Common Errors

The formula requires a function that is differentiable at the chosen base point. If \(f'(a)\) does not exist, there is no tangent slope supplied by the derivative definition, so this tangent-based construction is not available there. Also, \(a\) must be an input where the function is defined. It is not enough to substitute a number into the derivative formula if the original function is not defined at that number.

A common algebra error is to treat \(f'(a)\) as the line’s intercept. For the quadratic example above, a wrong calculation might be

This uses the derivative value \(5\) as the slope and the function value \(3\) as an added constant, but it does not use the required horizontal displacement \(x-a\). The check exposes the problem: \(L_{\text{wrong}}(2)=5(2)+3=13\), whereas \(f(2)=3\). The correct point-slope calculation is

Always retain the factor \(x-a\) until the point-slope expression is correctly formed. Expanding too early or confusing the point’s coordinates can produce a line with the right slope but the wrong height.

Why the Formula Matters

The linearization formula turns local information about a function into a concrete line. Once \(f(a)\) and \(f'(a)\) are known, the line is fixed: there is only one line through \((a,f(a))\) with slope \(f'(a)\). The formula makes that uniqueness visible and gives a repeatable way to construct the line without guessing its intercept.

It also keeps exact and approximate statements separate. At the base point, \(L(a)=f(a)\) exactly, and the tangent slope agrees with the derivative there. At other inputs, \(L(x)\) is a linear model, not a claim that \(L(x)=f(x)\). Differentiability explains why that model captures the function’s first-order change close to the base point, while the worked comparison shows that a difference can remain.