Linearizations of a function are lines—usually lines that can be used for purposes of calculation. Linearization is an effective method for approximating the output of a function {\displaystyle y=f(x)} at any {\displaystyle x=a} based on the value and slope of the function at {\displaystyle x=b}, given that {\displaystyle f(x)} is differentiable on {\displaystyle [a,b]} (or {\displaystyle [b,a]}) and that {\displaystyle a} is close to {\displaystyle b}. In short, linearization approximates the output of a function near {\displaystyle x=a}.

For example, {\displaystyle {\sqrt {4}}=2}. However, what would be a good approximation of {\displaystyle {\sqrt {4.001}}={\sqrt {4+.001}}}?

For any given function {\displaystyle y=f(x)}, {\displaystyle f(x)} can be approximated if it is near a known differentiable point. The most basic requisite is that {\displaystyle L_{a}(a)=f(a)}, where {\displaystyle L_{a}(x)} is the linearization of {\displaystyle f(x)} at {\displaystyle x=a}. The point-slope form of an equation forms an equation of a line, given a point {\displaystyle (H,K)} and slope {\displaystyle M}. The general form of this equation is: {\displaystyle y-K=M(x-H)}.

Using the point {\displaystyle (a,f(a))}, {\displaystyle L_{a}(x)} becomes {\displaystyle y=f(a)+M(x-a)}. Because differentiable functions are locally linear, the best slope to substitute in would be the slope of the line tangent to {\displaystyle f(x)} at {\displaystyle x=a}.

While the concept of local linearity applies the most to points arbitrarily close to {\displaystyle x=a}, those relatively close work relatively well for linear approximations. The slope {\displaystyle M} should be, most accurately, the slope of the tangent line at {\displaystyle x=a}.

An approximation of f(x)=x^2 at (xf(x))

Visually, the accompanying diagram shows the tangent line of {\displaystyle f(x)} at {\displaystyle x}. At {\displaystyle f(x+h)}, where {\displaystyle h} is any small positive or negative value, {\displaystyle f(x+h)} is very nearly the value of the tangent line at the point {\displaystyle (x+h,L(x+h))}.

The final equation for the linearization of a function at {\displaystyle x=a} is:

{\displaystyle y=(f(a)+f'(a)(x-a))}

For {\displaystyle x=a}, {\displaystyle f(a)=f(x)}. The derivative of {\displaystyle f(x)} is {\displaystyle f'(x)}, and the slope of {\displaystyle f(x)} at {\displaystyle a} is {\displaystyle f'(a)}.

a multivariable function