Find the third derivative.

*y* = *x*5 − 13*x*3 + 5

*y'''* =

To find the third derivative of the given function:

\[
y = x^5 - 13x^3 + 5
\]

### Step 1: Find the first derivative

\[
y' = \frac{d}{dx}(x^5 - 13x^3 + 5)
\]
Applying the power rule:

\[
y' = 5x^4 - 39x^2
\]

### Step 2: Find the second derivative

\[
y'' = \frac{d}{dx}(5x^4 - 39x^2)
\]
Again, applying the power rule:

\[
y'' = 20x^3 - 78x
\]

### Step 3: Find the third derivative

\[
y''' = \frac{d}{dx}(20x^3 - 78x)
\]

Applying the power rule one more time:

\[
y''' = 60x^2 - 78
\]

Thus, the third derivative is:

\[
y''' = 60x^2 - 78
\]

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Would you like more details or have any questions about the process?

### Related Questions:
1. What is the fourth derivative of the function?
2. How would you find the derivative of a trigonometric function?
3. How does the power rule apply to fractional exponents?
4. Can you explain the chain rule for more complex functions?
5. How do second and third derivatives relate to concavity and inflection points?

**Tip:** Always double-check exponents and coefficients while applying the power rule to avoid mistakes.

Find the third derivative.

y = x^5 − 13x^3 + 5
y''' =

Learn how to find the third derivative of the function y = x^5 - 13x^3 + 5 using the power rule. This detailed step-by-step solution covers first, second, and third derivatives, and is suitable for high school or college calculus students.

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