What am I doing wrong? Differentiation.

Question

I was given the following function:

$$ f(t) = \frac{t}{t^2 + 1} $$

And I attempted to derive it and came up with this answer:

$$ \frac{-2t^2}{(t^2 + 1)^2} $$

So I was incorrect obviously with this answer, and I am unsure now if you can obtain the correct answer using the chain rule. Perhaps I am just applying it incorrectly? I am able to get the correct result using the quotient rule.

I was under the impression that I could rearrange the function so that it is in the form $$f(g(x))$$ as follows: $$ f(t) = t (t^2 + 1)^{-1} $$

Answer 1

$t (t^2 + 1)^{-1}$ is not of the form $f(g(x))$.

$t (t^2 + 1)^{-1}$ is multiplication, it is $t \cdot (t^2 + 1)^{-1}$, we multiply $t$ by $(t^2 + 1)^{-1}$

$f(g(x))$ is function composition, it is $(f\circ g)(x)$, it is the function $f$ evaluated at $g(x)$.

Answer 2

You have to use the product rule as well as the chain rule. Write $f(t) = g(t) \cdot h(t)$ with $g(t) = t$ and $h(t) = (t^2 + 1)^{-1}$. Then

$$f'(t) = g'(t) \cdot h(t) + g(t) \cdot h'(t) = 1 \cdot (t^2 + 1)^{-1} - t \cdot 2t \cdot (t^2 + 1)^{-2} = \frac{t^2 + 1 - 2t^2}{(t^2 + 1)^2} = \frac{-t^2 + 1}{(t^2 + 1)^2}.$$