Difference Quotient Formula

The Difference Quotient Formula is a part of the definition of a function derivative. One can get derivative of a function by applying Limit h tends to zero i.e., h ⇢ 0 on difference quotient function. The difference quotient formula gives the slope of the secant line. A secant line is a line that passes through the two points of a curve. 

Let’s consider the curve y = f(x) and the secant line that passes through the two points are (x, f(x)) and (x + h, f(x+h)) then the difference quotient formula is given by-

Formula

Components of the formula:

1. 𝑓(𝑥):

This is the value of the function f at the point x.

2. f(x+h):

This is the value of the function f at the point x+h, where h is a small increment added to 𝑥

3. h:

This represents a small change in 𝑥. As h approaches zero, the difference quotient approaches the derivative of the functions.

Difference Quotient Formula Proof

Let’s consider the curve y = f(x) and the secant line that passes through the two points are (x, f(x)) and (x + h, f(x + h)).

Given,

(x₁, y₁) = (x, f(x))

(x₂, y₂) = (x + h, f(x + h))

Find the slope of the secant line,

Slope = (y₂ – y₁)/(x₂ – x₁) 

= (f(x + h) – f(x))/(x + h – x)

= (f(x + h) – f(x))/h

So the different quotient formula is slope of the secant line that passes through the given points.

Related Articles:

• Understanding Derivatives and Their Applications
• Fundamentals of Calculus: Limits and Continuity
• Applications of the Secant and Tangent Lines
• Introduction to Differential Calculus

Sample Problems

Below are a few sample questions on the Difference Quotient Formula that covers major types of problems.

Question 1: What is the difference quotient formula for the function f(x) = 7x + 9.

Solution:

Given,

f(x) = 7x + 9

Difference quotient formula = (f(x + h) – f(x))/h

= ((7(x + h) + 9) – (7x + 9))/h

= (7x + 7h + 9 – 7x – 9)/h

= 7h/h

= 7

Difference quotient formula for the given function is 7.

Question 2: What is the difference quotient formula for the function f(x) = 7x² – 1.

Solution:

Given,

f(x) = 7x² – 1

Difference quotient formula = (f(x + h) – f(x))/h

= ((7(x + h)² – 1) – (7x² – 1))/h

= ((7(x² + h² + 2xh) – 1) – (7x² – 1))/h

= (7x² + 7h² + 14xh – 1 – 7x² + 1)/h

= (7h² + 14xh)/h

= h(7h + 14x)/h

= 7h + 14x

Difference quotient formula for the given function is 7h + 14x.

Question 3: What is the difference quotient formula for the function f(x) = 25x

Solution:

Given,

f(x) = 25x

Difference quotient formula = (f(x + h) – f(x))/h

= ((25(x + h)) – (25x))/h

= (25x + 25h – 25x))/h

= 25h/h

= 25

Difference quotient formula for the given function is 25.

Question 4: What is the difference quotient formula for the function f(x) = √(x – 2)

Solution:

Given,

f(x) = √(x – 2)

Difference quotient formula = (f(x + h) – f(x))/h

= (√(x + h – 2) – √(x – 2))/h

[Tex]=\frac{\sqrt{x+h-2}-\sqrt{x-2}}{h}\times\frac{\sqrt{x+h-2}+\sqrt{x-2}}{\sqrt{x+h-2}+\sqrt{x-2}} =\frac{\sqrt{x+h-2}^2-\sqrt{x-2}^2}{h(\sqrt{x+h-2}+\sqrt{x-2})} =\frac{x+h-2-x+2}{h(\sqrt{x+h-2}+\sqrt{x-2})} =\frac{h}{h(\sqrt{x+h-2}+\sqrt{x-2})} =\frac{1}{\sqrt{x+h-2}+\sqrt{x-2}} [/Tex]

Difference quotient formula for the given function is 1/(√(x + h – 2) + √(x – 2)).

Question 5: What is the difference quotient formula for the function f(x) = 1/x.

Solution:

Given,

f(x) = 1/x

Difference quotient formula = (f(x + h) – f(x))/h

[Tex]=\frac{\frac{1}{x+h}-\frac{1}{x}}{h} =\frac{x-(x+h)}{h(x)(x+h))} =\frac{x-x-h}{h(x)(x+h)} =\frac{-h}{h(x)(x+h)} =\frac{-1}{(x)(x+h)} [/Tex]

Difference quotient formula for the given function is -1/(x)(x + h)

Question 6: Find difference Quotient for the function f(x) = 2x – 1

Solution:

Given f(x) = 2x – 1

Difference quotient = (f(x + h) – f(x))/h

= (2(x + h) – 1 – (2x – 1))/h

= (2x + 2h – 1 – 2x + 1)/h

= 2h/h

= 2

Hence Difference quotient for the function 2x – 1 is 2.

Question 7: What is the difference quotient for the function f(x) = log(x)

Solution:

Given f(x) = log(x)

Difference Quotient = (f(x + h) – f(x))/h

= (log(x + h) – log(x))/h

From Quotient property of logarithms log(a) – log(b) = log(a/b)

= log((x + h)/x)/h

So the difference quotient for the given function is log((x + h)/x)/h

Practice Problems on Difference Quotient Formula

1. Find the difference quotient for f(x)=x²

2. Determine the difference quotient for f(x)=sin(x).

3. Calculate the difference quotient for f(x)=e^x

4. Evaluate the difference quotient for f(x)=ln(3x).

5. Find the derivative of f(x)=x³+2x using the difference quotient formula.

6. Use the difference quotient to find the derivative of f(x)=cos(x).

7. Determine the difference quotient for f(x)= [Tex]root[/Tex] x

8. Find the difference quotient for f(x)=1/2x.

9. Calculate the difference quotient for f(x)=x⁴.

10. Use the difference quotient to determine the slope of the secant line for f(x)=tan(x).

Conclusion

The difference quotient formula is basic concept in calculus, which is used to calculate the slope of the secant line and ultimately the derivative of a function. With help of practice problems, you can gain a deeper clarity and understanding of how this formula is applied to various functions.

FAQs on Difference Quotient Formula

What is difference quotient formula?

difference quotient formula is used to calculate the slope of the secant line and ultimately the derivative of a function.

What is H in difference quotient formula?

h represents a small change or increment in the variable 𝑥

What is the quotient formula?

The quotient formula for the difference quotient is given by (f(x+h)-f(x))/h which approximates the slope of the secant line between two points on the function 𝑓(𝑥)

What is quotient Rule for difference?

The Quotient Rule for differences is a method to find the difference of two ratios, given by (ad-bc)/bd

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