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) = x^{2} + 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)
= x^{2} + 2xh + h^{2} + 2x + 2h  Step  2: Compute the difference f(x + h)  f(x).
f(x + h)  f(x) = [x^{2} + 2xh + h^{2} + 2x + 2h]  [x^{2} + 2x]
= x^{2} + 2xh + h^{2} + 2x + 2h  x^{2}  2x
= 2xh + h^{2} + 2h  Step  3: Divide the difference from Step  2 by h.
[f(x + h)  f(x)]/h = (2xh + h^{2} + 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:
Examples on F(x+h)F(x)/h

Example 1: Compute f(x+h)f(x)/h when f(x) = 2x  7.
Solution:
It is given that f(x) = 2x + 7.
Then f(x+h) = 2(x+h) + 7 = 2x + 2h + 7
Now, f(x+h)f(x)/h = [2x + 2h + 7  2x  7] / h
= 2h/h
= 2
Answer: f(x+h)f(x)/h = 2.

Example 2: Find the difference quotient of the function f(x) = x^{3}  4.
Solution:
It is given that f(x) = x^{3}  4.
Then f(x+h) = (x+h)^{3}  4 = x^{3} + 3x^{2}h + 3xh^{2} + h^{3}  4
The difference quotient of y = f(x) is:
f(x+h)f(x)/h = [x^{3} + 3x^{2}h + 3xh^{2} + h^{3 }  4  x^{3} + 4] / h
= [3x^{2}h + 3xh^{2} + h^{3}] / h
= h [3x^{2} + 3xh + h^{2}] / h
= 3x^{2} + 3xh + h^{2}
Answer: The difference quotient is 3x^{2} + 3xh + h^{2}.

Example 3: Find the average rate of change of the function f(x) = e^{x} over the interval [x, x + h].
Solution:
It is given that f(x) = e^{x}.
Then f(x+h) = e^{ (x + h)}.
The average rate of change of the function f(x) is:
f(x+h)f(x)/h = [e^{ (x + h)}  e^{x}]/h
= [e^{x} e^{h}  e^{x}]/h
= e^{x} [e^{h}  1] /h
Answer: The average rate of change is e^{x} [e^{h}  1] /h.
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 and it represents the slope of the secant line of the curve y = f(x).
How to Find F(x+h)F(x)/h?
To find f(x+h)f(x)/h for a given function y = f(x):
 Find f(x+h) by substituting x to be x + h in f(x).
 Find the difference f(x+h)f(x).
 Divide the above difference by h.
What is Another Name for F(x+h)F(x)/h?
f(x+h)f(x)/h is considered to be:
 the slope of the secant line
 difference quotient
 the average rate of change
What is the Limit of F(x+h)F(x)/h as h Tends to 0?
f(x+h)f(x)/h is the slope of the secant line passing through the points (x, f(x)) and (x+h, f(x+h)) on the curve y = f(x). When h tends to zero, the secant line becomes the tangent and in that case, f(x+h)f(x)/h represents the slope of the tangent and hence it also represents the derivative f '(x).
How is F(x+h)F(x)/h Related to Derivative of F(x)?
f(x+h)f(x)/h is involved in the limit definition of the derivative. For any function f(x), its derivative is f '(x) = lim ₕ → ₀ f(x+h)f(x)/h.
Is F(x+h)F(x)/h the Slope of Tangent Line?
No, f(x+h)f(x)/h is the slope of the secant line. It represents the slope of the tangent line only when h tends to 0.
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