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To find \(\frac{ds}{dt}\), we need to differentiate the function \( s \) with respect to \( t \). Given:

\[
s = 5t^5 - \sec t + 3e^t
\]

Let's differentiate each term separately:

1. **Differentiate \(5t^5\):**
   \[
   \frac{d}{dt}(5t^5) = 5 \cdot 5t^{4} = 25t^4
   \]

2. **Differentiate \(-\sec t\):**
   \[
   \frac{d}{dt}(-\sec t) = -\sec t \tan t
   \]

3. **Differentiate \(3e^t\):**
   \[
   \frac{d}{dt}(3e^t) = 3e^t
   \]

Now, combine all these results to find \(\frac{ds}{dt}\):

\[
\frac{ds}{dt} = 25t^4 - \sec t \tan t + 3e^t
\]

Therefore, the derivative \(\frac{ds}{dt}\) is:

\[
\frac{ds}{dt} = 25t^4 - \sec t \tan t + 3e^t
\]

Would you like more details or have any questions?

Here are some related questions that expand on this topic:

1. What are the rules for differentiating trigonometric functions?
2. How do you find higher-order derivatives of a function?
3. What is the geometric interpretation of the derivative?
4. How do you apply the chain rule in differentiation?
5. What are some applications of derivatives in real-world problems?

**Tip:** Always break down the differentiation process into smaller steps, especially when dealing with complex functions involving multiple terms.

Find ds/dt where s = 5t^5 - sec(t) + 3e^t.

Learn how to differentiate the function s = 5t^5 - sec(t) + 3e^t step by step. This problem covers key calculus concepts like the power rule, derivatives of exponential functions, and trigonometric functions. Suitable for advanced high school or college students.

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