Question

Finding a Derivative In Exercises \(7-34,\) find the derivative of the function. $$ g(t)=\frac{1}{\sqrt{t^{2}-2}} $$

Short answer

The derivative of the function \( g(t) = \frac{1}{\sqrt{t^{2}-2}} \) is \( g'(t) = -\frac{t}{t^{2}-2} \).

Step-by-step solution

Identify the Outer Function

The outer function can be seen as \( g(t) = \frac{1}{f(t)} \) where \( f(t) = \sqrt{t^{2}-2} \). So, to begin with, the power rule for differentiation has to be applied.

Apply the Power Rule

The power rule states that if \( g(t) = \frac{1}{f(t)} \) then its derivative \( g'(t) = -\frac{f'(t)}{[f(t)]^{2}} \). So now we need to find \( f'(t) \).

Find \( f'(t) \)

The function \( f(t) = \sqrt{t^{2}-2} \) can be rewritten as \( f(t) = (t^{2}-2)^{\frac{1}{2}} \). Now we can apply the chain rule to find \( f'(t) \).

Apply the Chain Rule

The chain rule states: If \( f(t) = (u(t))^n \), then \( f'(t) = n*(u(t))^{n-1}*u'(t) \). Here, \( u(t) = t^{2}-2 \), and it's derivative \( u'(t) = 2t \). Applying the chain rule gives \( f'(t) = \frac{1}{2}*(t^{2}-2)^{-\frac{1}{2}}*2t = \frac{t}{\sqrt{t^{2}-2}} \).

Find \( g'(t) \)

Substituting \( f(t) \) and \( f'(t) \) back into the equation for \( g'(t) \) gives \( g'(t) = -\frac{f'(t)}{[f(t)]^{2}} = -\frac{t/\sqrt{t^{2}-2}}{(\sqrt{t^{2}-2})^{2}} = -\frac{t}{t^{2}-2} \).

Key concepts

Power Rule

The power rule is a fundamental technique used in differentiation. It makes finding the derivative of polynomial expressions straightforward. When dealing with functions like \( g(t) = rac{1}{f(t)} \), we can simplify our work using the power rule.

• Given a function in the form \( x^n \), the power rule states that its derivative is \( nx^{n-1} \).
• In this context, \( f(t) = rac{1}{(t^2 - 2)^{1/2}} \) resembles \( x^{-n} \).

This means the derivative can be found using \( -f'(t)/(f(t))^2 \), representing \( n = -1 \). This simplification allows derivatives to be calculated more easily.

Chain Rule

The chain rule is used to differentiate composite functions. It helps when you have an "inner function" and an "outer function."

Consider a function \( f(x) = (u(x))^n \). Here, \( u(x) \) is the inner function and \( n \)is the power to which \( u(x) \) is raised. The chain rule formula states:

• Derivative: \( f'(x) = n \, (u(x))^{n-1} \, u'(x) \).

In our example, \( f(t) = (t^2 - 2)^{1/2} \). It means \( u(t) = t^2 - 2 \) with a derivative \( u'(t) = 2t \). Applying the chain rule efficiently lets us find \( f'(t) \):

• Calculate inner function's derivative \( u'(t) \).
• Combine with \( n and (u(t))^{n-1} \) terms to get the result.

Differentiation

Differentiation is the process of finding a derivative. It shows how a function changes at any given point. By differentiating, we determine the rate of change, or the slope, of the function. In our exercise, we're finding the derivative of \( g(t) = \frac{1}{\sqrt{t^{2}-2}} \). Here's what we do:

• First, identify any outer functions or combinations which need separation for calculus techniques, like the power rule and chain rule.
• Next, compute derivatives of any inner components using rules like the chain rule.
• Finally, substitute computed derivatives back into the original composite structure to find \( g'(t) \).

Outer Function

The concept of outer function is crucial in composite differentiation. It refers to the main function structure that encapsulates other "inner" functions. For example, in\( g(t) = \frac{1}{f(t)} \), the outer function is the fraction itself, \( g(t) \). This is what's directly differentiated using the power rule.

• Identify this structure before differentiating; it guides which derivative rules to apply.
• The outer function often dictates the initial differentiation steps, after which inner functions are differentiated separately and combined.
• Understanding this layer helps streamline the differentiation for complex expressions efficiently.