Use the Quotient Rule to differentiate the function.
f(t) = cos(t)/t^5



To differentiate the function \( f(t) = \frac{\cos(t)}{t^5} \) using the Quotient Rule, recall the rule:

\[
\frac{d}{dt} \left( \frac{u(t)}{v(t)} \right) = \frac{v(t) \cdot u'(t) - u(t) \cdot v'(t)}{v(t)^2}
\]

Here, \( u(t) = \cos(t) \) and \( v(t) = t^5 \). Now, differentiate \( u(t) \) and \( v(t) \):

- \( u'(t) = -\sin(t) \)
- \( v'(t) = 5t^4 \)

Now, apply the Quotient Rule:

\[
f'(t) = \frac{t^5 \cdot (-\sin(t)) - \cos(t) \cdot (5t^4)}{(t^5)^2}
\]

Simplify the numerator and denominator:

\[
f'(t) = \frac{-t^5 \sin(t) - 5t^4 \cos(t)}{t^{10}}
\]

Factor the numerator:

\[
f'(t) = \frac{-t^4 (t \sin(t) + 5 \cos(t))}{t^{10}}
\]

Finally, simplify:

\[
f'(t) = \frac{-(t \sin(t) + 5 \cos(t))}{t^6}
\]

So, the derivative is:

\[
f'(t) = \frac{-(t \sin(t) + 5 \cos(t))}{t^6}
\]

Would you like further clarification or details on any step? Here are 5 related questions:

1. What are the steps for applying the Product Rule in a similar function?
2. How does the Power Rule apply to the differentiation of \( t^5 \)?
3. What is the significance of factoring in the numerator when using the Quotient Rule?
4. Can you differentiate \( \sin(t)/t^5 \) using a similar process?
5. How would the derivative change if the function were \( \frac{1}{t^5} \)?

**Tip:** Always simplify expressions as much as possible after applying differentiation rules to make them easier to interpret.

Use the Quotient Rule to differentiate the function. f(t) = cos(t)/t^5

Learn how to differentiate the function f(t) = cos(t)/t^5 using the Quotient Rule. This calculus problem involves applying the Quotient Rule and simplifying the result step-by-step. Suitable for students in Grades 11-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation