Question

In Exercises \(75-80\), evaluate the derivative of the function at the indicated point. Use a graphing utility to verify your result. \(\frac{\text { Function }}{s(t)=\sqrt{t^{2}+2 t+8}} \quad \frac{\text { Point }}{(2,4)}\)

Short answer

The derivative of the function at the point (2,4) is \( \sqrt{3}/2 \).

Step-by-step solution

Rewrite the Function

In order to simplify the process of taking the derivative, the function is rewritten in exponential form, i.e. \(s(t) = (t^2 + 2t + 8)^{0.5}\). The exponent of 0.5 is equal to taking the square root.

Apply the Chain Rule

The function requires the application of the chain rule to evaluate its derivative, \(s'(t)\). The chain rule involves finding the derivative of the outer function and then multiplying it by the derivative of the inner function. Thus, \(s'(t) = 0.5*\frac{1}{(t^2 + 2t + 8)^{0.5}} * (2t+2)\).

Simplify the Expression

Simplify the derivative expression resulted in Step 2, which gives us \(s'(t) = (t+1)\ /\ (\sqrt{t^2+2t+8}\)

Evaluate the Derivative at Point

Now that we've found the derivative, the next step is to evaluate it at the point \((2,4)\). Just plug \(t = 2\) into the derivative expression from step 3, we get \(s'(2) = (2+1) /\ (\sqrt{2^2+2*2+8}) = 3/\sqrt{12} = \sqrt{3}/2.\)