Questions: Find the derivative of the function s(t) = 5/t - 5/t^2 + 5/t^3. (Express numbers in exact form. Use symbolic notation and fractions where needed.) s'(t) =

$Find the derivative of the function s(t) = 5/t - 5/t^2 + 5/t^3. (Express numbers in exact form. Use symbolic notation and fractions where needed.) s'(t) =$

Transcript text: Find the derivative of the function $s(t)=\frac{5}{t}-\frac{5}{t^{2}}+\frac{5}{t^{3}}$. (Express numbers in exact form. Use symbolic notation and fractions where needed.) \[ s^{\prime}(t)= \]

Solution

Solution Steps

Step 1: Rewrite the function in a form suitable for differentiation

Rewrite the function $ s(t) = \frac{5}{t} - \frac{5}{t^{2}} + \frac{5}{t^{3}} $ using negative exponents: \[ s(t) = 5t^{-1} - 5t^{-2} + 5t^{-3}. \]

Step 2: Apply the power rule to each term

Differentiate each term using the power rule $ \frac{d}{dt}[t^n] = n t^{n-1} $: \[ \frac{d}{dt}[5t^{-1}] = 5(-1)t^{-2} = -5t^{-2}, \] \[ \frac{d}{dt}[-5t^{-2}] = -5(-2)t^{-3} = 10t^{-3}, \] \[ \frac{d}{dt}[5t^{-3}] = 5(-3)t^{-4} = -15t^{-4}. \]

Step 3: Combine the derivatives

Combine the derivatives of each term to find $ s'(t) $: \[ s'(t) = -5t^{-2} + 10t^{-3} - 15t^{-4}. \]

Step 4: Rewrite the derivative in fractional form

Express the derivative using fractions: \[ s'(t) = -\frac{5}{t^{2}} + \frac{10}{t^{3}} - \frac{15}{t^{4}}. \]