F(x)/h - Formula, Derivation | Difference Quotient - Cuemath

F(x+h) - F(x)/h

f(x+h)-f(x)/h is a formula that is a part of limit definition of the derivative (first principles). The limit definition of the derivative of a function f(x) is, f'(x) = lim ₕ → ₀ [ f(x + h) - f(x) ] / h. But what is the connection between the derivative and f(x+h)-f(x)/h formula? Let us see.

Let us learn more about f(x+h)-f(x)/h along with its meaning, derivation, and examples.

1.	What is F(x+h) - F(x)/h?
2.	F(x+h) - F(x)/h Derivation
3.	How to Find F(x+h)-F(x)/h?
4.	FAQs on F(x+h)-F(x)/h

What is F(x+h) - F(x)/h?

f(x+h)-f(x)/h is called the difference quotient of a function f(x). What is the difference quotient of a function? Here, the words "difference" and "quotient" are giving a sense of the fraction of difference of coordinates and hence it represents the slope of a line that passes through two points of the curve. A line that intersects the curve at two points is called a secant line. Hence f(x+h)-f(x)/h represents the slope of the secant line.

F(x+h)-F(x)/h Formulas

Here are some formulas that are related to f(x+h)-f(x)/h:

• f(x+h)-f(x)/h is called the difference quotient of a function f(x).
• f(x+h)-f(x)/h is the slope of the secant line of a function f(x) passing through two points (x, f(x)) and (x + h, f(x + h)).
• f(x+h)-f(x)/h is the average of change of the function f(x) over the interval [x, x + h].
• lim ₕ → ₀ f(x+h)-f(x)/h gives the derivative of the function f(x) and is denoted by f '(x).

F(x+h) - F(x)/h Derivation

Consider the above figure where y = f(x) is a curve with two points A (x, f(x)) and B (x + h, f(x + h)) on it. Let us find the slope of the secant line AB using the slope formula. For this assume that A (x, f(x)) = (x₁, y₁) and B (x + h, f(x + h)) = (x₂, y₂). Then the slope of the secant line AB is,

(y₂ - y₁) / (x₂ - x₁)

= (f (x + h) - f(x)) / (x + h - x)

= f(x+h)-f(x)/h

Hence the formula.

How is F(x+h) - F(x)/h Connected to Derivative?

We know that the derivative of a function f(x) is nothing but the slope of the tangent. In the above figure, if point B approaches A (i.e., if B approximately coincides with A), then the secant line AB becomes the tangent line at A. For this, the horizontal distance between the two points A and B should be approximately 0. i.e., the secant line becomes the tangent line if h → 0. i.e.,

Slope of the tangent line = lim ₕ → ₀ f(x+h)-f(x)/h

(or)

f '(x) = lim ₕ → ₀ f(x+h)-f(x)/h

How to Find F(x+h)-F(x)/h?

Here are the steps to compute f(x+h)-f(x)/h for a given function f(x). The steps are explained through an example f(x) = x2 + 2x.

• Step - 1: Compute f(x + h) by substituting x = x + h on both sides of f(x). Then f(x + h) = (x + h)2 + 2(x + h) = x2 + 2xh + h2 + 2x + 2h
• Step - 2: Compute the difference f(x + h) - f(x). f(x + h) - f(x) = [x2 + 2xh + h2 + 2x + 2h] - [x2 + 2x] = x2 + 2xh + h2 + 2x + 2h - x2 - 2x = 2xh + h2 + 2h
• Step - 3: Divide the difference from Step - 2 by h. [f(x + h) - f(x)]/h = (2xh + h2 + 2h) / h = h (2x + h + 2) / h = 2x + h + 2

Important Points on F(x+h)-F(x)/h:

• f(x+h)-f(x)/h is called the formula of difference quotient.
• f(x+h)-f(x)/h gives the average rate of change of the function f(x) over the interval [x, x + h].
• f(x+h)-f(x)/h as h tends to 0 gives the derivative of f(x).
• f(a+h)-f(a)/h as h tends to 0 gives the slope of the tangent line of a curve y = f(x) at x = a.
• f(x+h)-f(x)/h for any line f(x) = mx + b is m.
• f(x+h)-f(x)/h for any constant function f(x) = c is 0.

Related Topics:

• f(x+h)-f(x)/h Calculator
• Derivative Calculator
• Tangent Line Calculator
• Antiderivative Calculator