Math Is Fun Forum

nycguitarguy

Member

Registered: 2024-02-24

Posts: 523

Difference Quotient of f

The slope of the secant line containing the two points
(x, f(x)) and (x + h, f(x + h)) on the graph of a function y = f(x) may be given as

m_sec = [f(x + h) - f(x)]/[(x + h) - x] which leads to [f(x + h) - f(x)]/h, where h cannot = 0.

Express the slope of the secant line for the function f(x) = 1/x^2 in terms of x and h. Be sure to simplify.

Let me see.

f(x) = [1/(x + h)^2 - (1/x^2)]/h

f(x) = [1/(x^2 + 2xh + h^2) - (1/x^2)]/h

After further simplifying, I got the following answer:

f(x) = (2x - h)/[(x^2)(x^2 + 2xh + h^2)]

You say?

Bob

Administrator

Registered: 2010-06-20

Posts: 10,184

Re: Difference Quotient of f

f(x) = (2x - h)/[(x^2)(x^2 + 2xh + h^2)]

Somewhere in that simplification you have lost a minus sign.  It should be

f'(x) = (-2x - h)/[(x^2)(x^2 + 2xh + h^2)]

Then when h tends to zero it becomes f'(x) = (-2x )/[(x^2)(x^2)] = -2/x^3

Other minor point. Don't start with the secant function as f(x). f(x) is the original function not the secant function.  I've used f'(x) as this is a common notation for the differential function.

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

nycguitarguy

Member

Registered: 2024-02-24

Posts: 523

Re: Difference Quotient of f

f(x) = (2x - h)/[(x^2)(x^2 + 2xh + h^2)]

Somewhere in that simplification you have lost a minus sign.  It should be

f'(x) = (-2x - h)/[(x^2)(x^2 + 2xh + h^2)]

Then when h tends to zero it becomes f'(x) = (-2x )/[(x^2)(x^2)] = -2/x^3

Other minor point. Don't start with the secant function as f(x). f(x) is the original function not the secant function.  I've used f'(x) as this is a common notation for the differential function.

Bob

Ok. Thanks for letting me know.
Now, does f '(x) mean f prime of x?

Is f prime of x the slope which is the derivative?

Bob

Administrator

Registered: 2010-06-20

Posts: 10,184

Re: Difference Quotient of f

yes.

There's two notations used but they mean the same f'(x) = dy/dx

Note. If you had z = f(x) then it would be dz/dx

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

nycguitarguy

Member

Registered: 2024-02-24

Posts: 523

Re: Difference Quotient of f

yes.

There's two notations used but they mean the same f'(x) = dy/dx

Note. If you had z = f(x) then it would be dz/dx

Bob

I can say dy/dx reads d y d x. YES?

If I have w = f(x), I can then say dw/dx. YES?

If I had f(z), the derivative is f prime of z. Yes?

Bob

Administrator

Registered: 2010-06-20

Posts: 10,184

Re: Difference Quotient of f

In the UK folk usually say d y by d x .  Your other statements are correct.

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

nycguitarguy

Member

Registered: 2024-02-24

Posts: 523

Re: Difference Quotient of f

In the UK folk usually say d y by d x .  Your other statements are correct.

Bob

Glad to know my statements are correct.