- #1

kingwinner

- 1,270

- 0

## Homework Statement

Are (d/dt)[f(x+t)] and f '(x+t) always equal?

## Homework Equations

## The Attempt at a Solution

For (d/dt)[f(x+t)], I would interpret it as first evaluate at x+t, then differentiate with respect to t.

And the standard interpretation of f '(x+t) is f '(x+t) = (d[f(x)]/dx) |x=x+t (first differenate, then evaluate at x+t, so the derivative itself is treated as a function).

I've tried a few functions, and they both expressions give the same answer which is quite surprising to me...

e.g.) f(x)=x^2, f '(x)=2x, f '(x+t)=2(x+t)

f(x+t)=(x+t)^2, (d/dt)[f(x+t)]=2(x+t)

e.g.) f(x)=ln(x), f '(x)=1/x, f '(x+t)=1/(x+t)

f(x+t)=ln(x+t), (d/dt)[f(x+t)]=1/(x+t)

However, these examples may just be special cases. It's just does not seem clear to me that these two expressions are always equal.

Is there any way to prove rigorously that these two expressions indeed always give the same answer? (or disprove it using a counterexample?)

Thanks for any help!