Problem 33 - College Algebra and Calculus: An Applied Approach

Question

Use the General Power Rule to find the derivative of the function. $$ s(t)=\sqrt{2 t^{2}+5 t+2} $$

Step-by-Step Solution

Answer

The derivative of the given function is \( s'(t) = (4t + 5) / (2\sqrt{2t^2 + 5t + 2}) \).

Step 1: Rewrite in Exponential Form

Rewrite the square root function in exponential form. The function thus becomes: \( s(t) = (2t^2 + 5t + 2)^{1/2} \).

Step 2: Apply the Chain Rule and Power Rule

Use the chain rule, which states that the derivative of a composite function is the derivative of the outside function, multiplied by the derivative of the inside function. Here, our outside function is \( u^{1/2} \) and our inside function is \( u = 2t^2 + 5t + 2 \). Apply the power rule to the outside function and differentiate the inside function to get: \( s'(t) = 1/2 (2t^2 + 5t + 2)^{-1/2} * (4t + 5) \).

Step 3: Simplify the Result

Simplify the derivative to its simplest form, which is: \( s'(t) = (4t + 5) / (2\sqrt{2t^2 + 5t + 2}) \).