# Linear Approximation (Calculus) The concept of **tangent line approximation** relies heavily on the point-slope formula. This application of the tangent line equation may not be extremely useful by itself, but it can assist in understanding more complicated topics such as [[Differentials]]. This technique is often useful when trying to find the derivative of complex functions that are difficult to calculate normally, such as: - Complex Roots: $f(x) = \sqrt{9.213}$ - Trigonometric Functions: $f(x) = \sin (62\degree)$ > [!summary] > Let $a$ be a value that is very close to $x$. > > $L(x) = f(a) + f'(a)(x-a)$ > > This is known as the **linear approximation** (or tangent line approximation) of the function $f$ at $x=a$. > > The function $L$ is also known as the **linearization** of $f$ at $x=a$ Linear approximations use a point, $(a, f(a))$, that is very close to the point $(x, f(x))$. The equation of the tangent line for the function $f(x)$ at the point $(a, f(a))$ can then be written as: