Derivative of a function can be used to find the linear approximation of a function at a given value. The linear approximation method was given by Newton and he suggested finding the value of the function at the given point and then finding the equation of the tangent line to find the approximately close value to the function. The equation of the function of the tangent is
L(x) = f(a) + f'(a)(x−a)
The tangent will be a very good approximation to the function’s graph and will give the closest value of the function. Let us understand this with an example, we can estimate the value of √9.1 using the linear approximation. Here we have the function: f(x) = y = √x. We will find the value of √9 and using linear approximation, we will find the value of √9.1.
We have f(x) = √x, then f'(x) = 1/(2√x)
Putting a = 9 in L(x) = f(a) + f'(a)(x−a), we get,
L(x) = f(9) + f'(9)(9.1−9)
L(x) = 3 + (1/6)0.1
L(x) ≈ 3.0167.
This value is very close to the actual value of √(9.1)
Hence by using derivatives, we can find the linear approximation of function to get the value near to the function.