Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

For example, $f(x) = 3x^2$, and I want to obtain $f '(2)$ using only information I have with the point $x = 1$. Is there an equation I could use to do this which would apply for all functions, not just the one I mentioned above?

share|improve this question
    
Why would you want to do this? You have $f$. What prevents you from simply taking the derivative? –  EuYu Oct 11 '12 at 19:55
    
I know I can do this, I was just wondering if there's a relationship between consecutive points involving higher derivatives (2nd derivative and so on). –  Adi Oct 11 '12 at 19:57
    
What do you mean by "consecutive points"? For the question you asked in the post, the answer is no. You cannot determine $f'(2)$ knowing only $f(1)$. You can't determine $f'(2)$ knowing every derivative of $f$ at 1. You cannot even determine $f'(2)$ knowing the value of $f(x)$ for all $x \in (-\infty, 3/2)$. If you assume $f$ is analytic or polynomial, things become more interesting. –  Hurkyl Oct 13 '12 at 15:31

1 Answer 1

A very elementary method would be to take the first difference of the equation $$\Delta f(x)=f(x+1) - f(x)$$ which would give you the secant through the two points. We can make a rough approximation that $$\frac{f'(x+1) + f'(x)}{2} \approx \Delta f(x)$$ which in turn gives $$f'(x+1) \approx 2\Delta f(x) - f'(x)$$ For this particular function, this works out quite well. It approximates $f'(2)$ perfectly.

I don't know how much background in calculus you have, so what follows may either be complete trivial to you or it might not.

The above method can be improved quite a bit by something called Taylor Polynomials (or more generally Taylor series). Any sufficiently smooth function can be closely approximated by a polynomial. The higher the degree of the polynomial, the smaller the error term. If we take our point of interest, say $a$, then we can write a Taylor polynomial for $f$ centered at $a$. In the case of your example, we can rewrite the function $f(x) = 3x^2$ centered at $x=1$ as $$f(x) = f(1) + f'(1)(x-1) + \frac{f''(1)}{2}(x-1)^2$$ You can differentiate this series to give $$f'(x) = f'(1) + f''(1)(x-1)$$ This is an expression for the derivative using only information from $x=1$. If you actually evaluate the expression, you will get $6x$ which is exactly as you would expect. Polynomials are a bit too simple too emphasize methods like this. In general things will not work out exactly as they have in this example.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.