To find the derivative of \( \frac{\cos(x)}{x^3} \), we will use the **quotient rule**. The quotient rule states that for two functions \( f(x) \) and \( g(x) \):

\[
\frac{d}{dx} \left( \frac{f(x)}{g(x)} \right) = \frac{g(x) f'(x) - f(x) g'(x)}{(g(x))^2}
\]

Here, let:

- \( f(x) = \cos(x) \), so \( f'(x) = -\sin(x) \),
- \( g(x) = x^3 \), so \( g'(x) = 3x^2 \).

Now applying the quotient rule:

\[
\frac{d}{dx} \left( \frac{\cos(x)}{x^3} \right) = \frac{x^3(-\sin(x)) - \cos(x)(3x^2)}{(x^3)^2}
\]

Simplifying:

\[
= \frac{-x^3 \sin(x) - 3x^2 \cos(x)}{x^6}
\]

Finally, simplify further by canceling \( x^2 \) from both the numerator and denominator:

\[
= \frac{-x \sin(x) - 3 \cos(x)}{x^4}
\]

So, the derivative is:

\[
\boxed{ \frac{-x \sin(x) - 3 \cos(x)}{x^4} }
\]

Would you like further details on this process or have any questions?

---

Here are some related questions:
1. How do you apply the product rule to similar problems?
2. What happens if you take the second derivative of this function?
3. How do you find critical points for this function?
4. Can you explain the chain rule in the context of trigonometric functions?
5. How does this derivative behave as \( x \to 0 \)?

**Tip:** Always simplify before applying the quotient rule to avoid unnecessary complexity.

Find the derivative of cos(x)/x^3 using the quotient rule. This problem explains key calculus concepts like the derivative of trigonometric functions and the quotient rule, with a detailed step-by-step solution. Suitable for college-level calculus students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation