Question

Estimate the indicated derivative by any method. (Round your answer to three decimal places.)

g(t) = 3/t⁵
Estimate g'(1).

Answer 1

The estimated value of
`\( g'(1) \)` for
`\( g(t) = (3)/(t^5) \)` is ( 15.000 ) (rounded to three decimal places).

Step-by-step explanation:

To estimate
`\( g'(1) \)`, we can use the definition of the derivative:

`\[ g'(t) = \lim_{{h \to 0}} (g(t + h) - g(t))/(h) \]`

Apply this formula to
`\( g(t) = (3)/(t^5) \)`:

`\[ g'(t) = \lim_{{h \to 0}} ((3)/((t + h)^5) - (3)/(t^5))/(h) \]`

Now, substitute ( t = 1 ) into this expression:

`\[ g'(1) = \lim_{{h \to 0}} ((3)/((1 + h)^5) - (3)/(1^5))/(h) \]`

Simplify further:

\
`[ g'(1) = \lim_{{h \to 0}} ((3)/((1 + h)^5) - 3)/(h) \]`

Now, compute the limit. As ( h ) approaches 0, the expression becomes
`\( (3 - 3)/(0) \)`, which is an indeterminate form. Apply L'Hôpital's Rule, taking the derivative of the numerator and denominator with respect to ( h ):

`\[ g'(1) = \lim_{{h \to 0}} ((d)/(dh)\left((3)/((1 + h)^5) - 3\right))/((d)/(dh)h) \]`

After evaluating this expression, we find (g'(1) = 15.000) (rounded to three decimal places).