Differential of a function

The concept of differential is linked to those of differential function and increase of the dependent variable, at a point in the domain of the derivative .

Summary

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• 1 Definition
• 2 Accuracies
• 3 Approximation of the differential
• 4 Properties
• 5 References and notes
• 6 See
• 7 Sources

Definition

Since y = f (x) is a differentiable function at point x, the differential of y (at the value xy for an increase Δ x) is expressed by

dy = f ‘(x) Δx, considering Δx an arbitrary increment of x. ^[one]

Accuracy

• you can write dy = df, dy = (dy ÷ dx) · dx
• The differential, strictly speaking, is a function of two variables of x and Δx, where x is a point in the domain of f ‘and Δx an arbitrary real number.
• Having to write properly

df = df (x, Δx) = f ‘(x) Δx

Differential approximation

For a small increase in Δx, the differential approaches the increase Δy == f (x + Δx) – f (x), it is true

Δy = dy

or otherwise

y + Δy = y + f ‘(x) Δx

Properties

1. The differential of the sum of two differentiable functions g and h is equal to the sum of the differentials of such functions: d (g + h) = dg + dh
2. For the product of functions, the equality d (gh) = gdh + hdg fits
3. When y = g / h, dy = (hdg-gdh) ÷ h ²

Examples

1. y = sec x, dy = secx tanx dx
2. s = (1 + ln t) ²implies ds = 2 (1 + ln t) · (1 / t) dt.
3. In the case of the compound function y = f [g (x)] we have dy = f ‘ _u(u) · u’ _x dx, where u = g (x)