We are asked to find the
third derivative of the functions in
Exercises 13 and 17, as specified.
---
##
✔ Exercise 13:
Given:
> \( f(x) = x^5 - 3x^4 \)
We need to compute \( f'''(x) \), the third derivative.
---
Step 1: First derivative
Use power rule: \( \frac{d}{dx} x^n = n x^{n-1} \)
\[
f'(x) = \frac{d}{dx}(x^5) - 3 \cdot \frac{d}{dx}(x^4) = 5x^4 - 12x^3
\]
---
Step 2: Second derivative
\[
f''(x) = \frac{d}{dx}(5x^4) - \frac{d}{dx}(12x^3) = 20x^3 - 36x^2
\]
---
Step 3: Third derivative
\[
f'''(x) = \frac{d}{dx}(20x^3) - \frac{d}{dx}(36x^2) = 60x^2 - 72x
\]
✔ Final Answer for #13:
\[
\boxed{f'''(x) = 60x^2 - 72x}
\]
---
##
✔ Exercise 17:
Given:
> \( f(x) = \frac{3}{8x^4} \)
We can rewrite this using negative exponents to make differentiation easier:
\[
f(x) = \frac{3}{8} \cdot x^{-4}
\]
Now compute the third derivative.
---
Step 1: First derivative
\[
f'(x) = \frac{3}{8} \cdot (-4) x^{-5} = -\frac{12}{8} x^{-5} = -\frac{3}{2} x^{-5}
\]
---
Step 2: Second derivative
\[
f''(x) = -\frac{3}{2} \cdot (-5) x^{-6} = \frac{15}{2} x^{-6}
\]
---
Step 3: Third derivative
\[
f'''(x) = \frac{15}{2} \cdot (-6) x^{-7} = -\frac{90}{2} x^{-7} = -45 x^{-7}
\]
We can write this back in fractional form if preferred:
\[
f'''(x) = -\frac{45}{x^7}
\]
✔ Final Answer for #17:
\[
\boxed{f'''(x) = -\frac{45}{x^7}}
\]
---
## 📝 Summary:
-
#13: \( \boxed{f'''(x) = 60x^2 - 72x} \)
-
#17: \( \boxed{f'''(x) = -\frac{45}{x^7}} \)
Let me know if you’d like to see steps for any other problems!
Parent Tip: Review the logic above to help your child master the concept of higher order derivatives worksheet.
