## Application of Derivatives

# Approximations

- Let y = f(x), Δx be a small increment in x and Δy be the increment in y corresponding to the increment in x, i.e., Δ
*y =*f(x*+ Δx*) − f(*x*). Then*dy*given by

dy=f'(x)dx\ or\ dy=\left(\frac{dy}{dx}\right)\Delta x.

is a good approximation of Δy when*dx*= Δx is relatively small and we denote it by*dy*≈ Δ*y*. - \frac{\Delta y}{y} is called Relative error in y

\frac{\Delta y}{y}\times 100is called percentage error in y.

### View the Topic in this video From 35:00 To 51:56

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- If y = f(x) be a function. Δx is small change in x, then the corresponding change in y is Δy, which is known as Approximate error in y.

Δy = f'(x) Δx = dy - If f(x) = ax
^{2}+ bx + c then Δf − df = a(Δx)^{2} - If y = f(x) is any function, then f(x + Δx) = f(x) + f'(x)Δx
- If y = f(x) is a homogeneous function of degree n (or) y = x
^{n}then

→ Relative error in y = n (relative error in x)

→ % error in y = n (% error in x)