Consider the following function.
g(t) = t

sqrt8 − t
,    t < 7
Find
g'(t).
g'(t) =

Excluding any values not in the domain of g, for what values of c is
g'(c) = 0
or undefined? (Enter your answers as a comma-separated list.)
c =

Give the critical numbers of the function. (Enter your answers as a comma-separated list.)
t =
.

To find the derivative \( g'(t) \) of the given function and determine the critical numbers, let's proceed step by step.

The function given is:

\[ g(t) = \frac{t}{\sqrt{8 - t}} \]

### Finding \( g'(t) \)

First, we can rewrite the function to make differentiation easier:

\[ g(t) = t (8 - t)^{-1/2} \]

Now we use the product rule to find the derivative:

\[ g(t) = u \cdot v \]
where:
\[ u = t \]
\[ v = (8 - t)^{-1/2} \]

The product rule states that:
\[ g'(t) = u'v + uv' \]

First, we need to find the derivatives of \( u \) and \( v \):

\[ u' = \frac{d}{dt}(t) = 1 \]
\[ v = (8 - t)^{-1/2} \]
\[ v' = \frac{d}{dt} (8 - t)^{-1/2} \]

Using the chain rule for \( v' \):

\[ v' = -\frac{1}{2} (8 - t)^{-3/2} \cdot (-1) = \frac{1}{2} (8 - t)^{-3/2} \]

Now, applying the product rule:

\[ g'(t) = u'v + uv' \]
\[ g'(t) = 1 \cdot (8 - t)^{-1/2} + t \cdot \frac{1}{2} (8 - t)^{-3/2} \]
\[ g'(t) = (8 - t)^{-1/2} + \frac{t}{2} (8 - t)^{-3/2} \]

Simplifying further:

\[ g'(t) = \frac{1}{\sqrt{8 - t}} + \frac{t}{2(8 - t

Learn how to find the derivative and critical numbers of the function g(t) = t / sqrt(8 - t). Understand the application of differentiation rules such as the product rule and chain rule. Discover for which values of t the derivative is 0 or undefined. Suitable for advanced high school or college-level mathematics.

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