Find the first and second derivatives. $$ s=\frac{t^{5}+3 t^{4}-4}{t^{5}} $$

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The first derivative is $$ \frac{ds}{dt} = -\frac{3}{t^{2}} + \frac{20}{t^{6}} $$ and the second derivative is $$ \frac{d^2s}{dt^2} = \frac{6}{t^{3}} - \frac{120}{t^{7}} $$. To find the first and second derivatives of the function $$ s = \frac{t^{5} + 3t^{4} - 4}{t^{5}}, $$ we can first simplify the expression. We can rewrite $$ s $$ as follows: $$ s = \frac{t^{5}}{t^{5}} + \frac{3t^{4}}{t^{5}} - \frac{4}{t^{5}} = 1 + \frac{3}{t} - \frac{4}{t^{5}}. $$ Now, we can differentiate $$ s $$ with respect to $$ t $$. ### First Derivative Using the power rule and the derivative of $$ t^{-n} $$: $$ s = 1 + 3t^{-1} - 4t^{-5}. $$ Now, we differentiate term by term: 1. The derivative of $$ 1 $$ is $$ 0 $$. 2. The derivative of $$ 3t^{-1} $$ is $$ -3t^{-2} $$. 3. The derivative of $$ -4t^{-5} $$ is $$ 20t^{-6} $$. Putting it all together, we have: $$ \frac{ds}{dt} = 0 - 3t^{-2} + 20t^{-6} = -\frac{3}{t^{2}} + \frac{20}{t^{6}}. $$ ### Second Derivative Now, we differentiate $$ \frac{ds}{dt} $$ to find the second derivative: $$ \frac{ds}{dt} = -3t^{-2} + 20t^{-6}. $$ Differentiating term by term again: 1. The derivative of $$ -3t^{-2} $$ is $$ 6t^{-3} $$. 2. The derivative of $$ 20t^{-6} $$ is $$ -120t^{-7} $$. Thus, the second derivative is: $$ \frac{d^2s}{dt^2} = 6t^{-3} - 120t^{-7} = \frac{6}{t^{3}} - \frac{120}{t^{7}}. $$ ### Summary The first and second derivatives of the function $$ s $$ are: $$ \frac{ds}{dt} = -\frac{3}{t^{2}} + \frac{20}{t^{6}}, $$ $$ \frac{d^2s}{dt^2} = \frac{6}{t^{3}} - \frac{120}{t^{7}}. $$

Find the first and second derivatives. \( s=\frac{t^{5}+3 t^{4}-4}{t^{5}} \)

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Find the first and second derivatives. \( s=\frac{t^{5}+3 t^{4}-4}{t^{5}} \)

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