find the derivative of f(t)=(t^4+t^5-1)/t^6 without using the quotient rule, simplify the equation first then find the derivative

To find the derivative of the function \( f(t) = \frac{t^4 + t^5 - 1}{t^6} \), we will first simplify the equation by dividing each term in the numerator by \( t^6 \), avoiding the use of the quotient rule.

### Step 1: Simplify the function
We can rewrite the function as follows:
\[
f(t) = \frac{t^4}{t^6} + \frac{t^5}{t^6} - \frac{1}{t^6}
\]

Simplifying each term:
\[
f(t) = t^{4-6} + t^{5-6} - t^{-6}
\]
\[
f(t) = t^{-2} + t^{-1} - t^{-6}
\]

### Step 2: Find the derivative
Now that we have simplified the function, we can find the derivative by applying the power rule to each term. The power rule states that the derivative of \( t^n \) is \( n \cdot t^{n-1} \).

For \( t^{-2} \):
\[
\frac{d}{dt} \left( t^{-2} \right) = -2t^{-3}
\]

For \( t^{-1} \):
\[
\frac{d}{dt} \left( t^{-1} \right) = -t^{-2}
\]

For \( t^{-6} \):
\[
\frac{d}{dt} \left( t^{-6} \right) = -6t^{-7}
\]

### Step 3: Combine the results
Now, combining all the derivatives:
\[
f'(t) = -2t^{-3} - t^{-2} + 6t^{-7}
\]

Thus, the derivative of \( f(t) \) is:
\[
f'(t) = -\frac{2}{t^3} - \frac{1}{t^2} + \frac{6}{t^7}
\]

Would you like a detailed explanation of any of these steps, or have any questions?

Here are 5 related questions you might find helpful:

1. How do you apply the quotient rule, and when is it useful?
2. What is the power rule, and why is it a fundamental tool in calculus?
3. How would you find the second derivative of \( f(t) \)?
4. What is the chain rule, and how does it differ from the power rule?
5. How can you interpret derivatives in a real-world context?

**Tip**: When simplifying fractions in calculus, reducing the terms first often leads to quicker and easier differentiation.

Find the derivative of f(t) = (t^4 + t^5 - 1) / t^6 without using the quotient rule, simplify the equation first then find the derivative.

Learn how to find the derivative of f(t) = (t^4 + t^5 - 1) / t^6 without using the quotient rule. This solution explains the simplification process using exponent rules, followed by applying the power rule to find the derivative. Suitable for high school and college calculus students.

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