Problem 57
Question
Find the derivative of the function. State which differentiation rule(s) you used to find the derivative. $$ g(t)=\frac{1}{t^{2}-2} $$
Step-by-Step Solution
Verified Answer
The derivative of the given function is \(g'(t) = -\frac{2t}{{(t^{2}-2)}^{2}}\). The Quotient Rule was used for differentiating.
1Step 1: 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 \(v = t^2 - 2\).
2Step 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 \(2t\).
3Step 3: 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.
Other exercises in this chapter
Problem 56
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