Question

Find the derivative of the function. State which differentiation rule(s) you used to find the derivative. $$ g(t)=\frac{1}{t^{2}-2} $$

Short answer

The derivative of the given function is \(g'(t) = -\frac{2t}{{(t^{2}-2)}^{2}}\). The Quotient Rule was used for differentiating.

Step-by-step solution

Identify the functions u and v

In the problem, \(g(t)=\frac{1}{t^{2}-2}\), which is of the form \(u/v\). Here, \(u = 1\) and \(v = t^2 - 2\).

Work out du/dt and dv/dt

The derivative of u with respect to t, du/dt, is the derivative of 1, which is 0. The derivative of v with respect to t, dv/dt, is the derivative of \(t^2 - 2\), which is \(2t\).

Apply the Quotient Rule

According to the Quotient Rule, the derivative of \(u/v\) is \(v\cdot du/dt - u\cdot dv/dt\) divided by \(v^2\). Substituting the given values, \(g'(t) = (t^{2}-2)*0-1*(2t)\)/(t^{2} - 2)^2 = -2t/(t^{2} - 2)^2.