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Approximations - Application of Derivatives

Last Updated : 19 Sep, 2025

In calculus, approximations in derivatives usually mean using the derivative to estimate how a function behaves near a point β€” instead of working with the exact (and often complicated) formula of the function.

Let f be a given function and let y = f(x). Let βˆ†x denote a small increment in x.

πŸ‘ Application-of-Derivatives-2

Now the increment in y is like the increment in x, denoted by 

βˆ†y, is given by βˆ†y = f (x + βˆ†x) - f (x)

β‡’ βˆ†y/βˆ†x = [f(x + βˆ†x) - f (x)]/βˆ†x

If dx = βˆ†x is relatively small when compared to with x dy β‰ˆ βˆ†y.

β‡’ dy/dx β‰ˆ [f(x + βˆ†x) - f (x)]/βˆ†x

β‡’ dy/dx Γ— βˆ†x + f(x) β‰ˆ f(x + βˆ†x)

Thus, f(x + βˆ†x) β‰ˆ f'(x) Γ— βˆ†x + f(x)

Therefore, the formula for linear approximation is given as:

f(x + βˆ†x) β‰ˆ f(x) + fβ€²(x)(βˆ†x)

Where:

  • f(a) is the value of the function at a,
  • fβ€²(a) is the derivative of the function at a,
  • (x βˆ’ a) is the deviation from point a.

Example 1: Find the approximate value of √26.

Solution:

Let the f(x) = √x and the derivative of this is f’(x)= 1/(2√x)

Now we know the formula of approximation

f(x + βˆ†x) β‰ˆ f(x) + fβ€²(x)(βˆ†x)

Here we will assume x near to 25 which is a perfect square.

So we will assume βˆ†x = 1

f(x + βˆ†x) = f(x) + f’(x) . βˆ†x
β‡’ f(25 + 1) = f(25) + f'(25) Γ— 1
β‡’ f(26) = √25 + (1/(2 Γ— √(25))
β‡’ f(26) =  5 + 1/10 √26 
β‡’ f(26) = 5 + 0.1  =  5.1 

Example 2: Find the approximate value of f(3.02), where f(x) = 3x2 + 5x + 3. 

Solution: 

Let x = 3 and Ξ”x = 0.02. Then,

Since, f(3.02) = f(x + Ξ”x) = 3(x + Ξ”x)2 + 5(x + Ξ”x) + 3

Note that Ξ”y = f(x+Ξ”x) - f(x). 

Therefore, f(x + Ξ”x) = f(x) + Ξ”y

β‰ˆ f(x) + f'(x)Ξ”x (as ds = Ξ”x)
β‡’ f(3.02)  β‰ˆ (3x2 + 5x + 3) + (6x + 5)Ξ”x
β‡’ f(3.02) = (27 + 15 + 3) + (18 + 5)(0.02)
β‡’ f(3.02) = 45 + 0.46 = 45.46

Hence, the approximate value of f(3.02) is 45.46.

Numerical Differentiation (Finite Differences)

In situations where the derivative of a function is difficult to compute analytically, numerical differentiation can be used to approximate derivatives. These approximations are based on finite differences.

  • Forward Difference: fβ€²(x) β‰ˆ [f(x + h) βˆ’ f(x)]/h
  • Central Difference: fβ€²(x) β‰ˆ [f(x + h) βˆ’ f(xβˆ’h)]/2h

Higher-Order Approximations

A higher-order approximation can be derived using the Taylor series, which expands a function into an infinite sum of terms based on the function’s derivatives at a specific point. The more terms we include, the better the approximation becomes.

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