Given some examples of solving question from Chain Rule. Follow the steps and solve rest of the questions given below.
Question 1: Differentiate \(f(x) = e^{4x^3 – 2x}\).
Step-by-step solution:
- Outer function: \(e^u\), where \(u = 4x^3 – 2x\).
- Differentiate outer: \(\frac{d}{du} e^u = e^u\).
- Differentiate inner: \(\frac{du}{dx} = 12x^2 – 2\).
- Apply chain rule: \(f'(x) = e^u \cdot \frac{du}{dx}\).
- Final answer: \(f'(x) = (12x^2 – 2)e^{4x^3 – 2x}\)
Question 2: Find \(f'(x)\) for \(f(x) = \left( \ln(5x + 3) \right)^4\).
Step-by-step Solution:
- Outer function: \(u^4\), where \(u = \ln(5x + 3)\).
- Differentiate outer: \(4u^3\).
- Differentiate inner: \(\frac{du}{dx} = \frac{5}{5x + 3}\).
- Apply chain rule: \(f'(x) = 4u^3 \cdot \frac{du}{dx}\).
- Final answer: \(f'(x) = \frac{20(\ln(5x + 3))^3}{5x + 3}\)
Question 3: Differentiate \(f(x) = \frac{1}{(3x^2 + 2x – 1)^5}\).
Step-by-step Solution:
- Rewrite: \(f(x) = (3x^2 + 2x – 1)^{-5}\).
- Outer function: \(u^{-5}\), where \(u = 3x^2 + 2x – 1\).
- Differentiate outer: \(-5u^{-6}\).
- Differentiate inner: \(\frac{du}{dx} = 6x + 2\).
- Apply chain rule: \(f'(x) = -5(3x^2 + 2x – 1)^{-6} \cdot (6x + 2) = \frac{-10(3x + 1)}{(3x^2 + 2x – 1)^6}\)
Question 4: Find \(f'(x)\) for \(f(x) = \sqrt{\ln(2x + 1)}\).
Step-by-step Solution:
- Rewrite: \(f(x) = (\ln(2x + 1))^{1/2}\).
- Outer function: \(u^{1/2}\), where \(u = \ln(2x + 1)\).
- Differentiate outer: \(\frac{1}{2}u^{-1/2}\).
- Differentiate inner: \(\frac{du}{dx} = \frac{2}{2x + 1}\).
- Apply chain rule: \(f'(x) = \frac{1}{(2x + 1)\sqrt{\ln(2x + 1)}}\)
Question 5: Differentiate \(f(x) = (e^{2x} + x^3)^{-3}\).
Step-by-step Solution:
- Outer function: \(u^{-3}\), where \(u = e^{2x} + x^3\).
- Differentiate outer: \(-3u^{-4}\).
- Differentiate inner: \(\frac{du}{dx} = 2e^{2x} + 3x^2\).
- Apply chain rule: \(f'(x) = -3(e^{2x} + x^3)^{-4} \cdot (2e^{2x} + 3x^2)\)
- Simplify: \(f'(x) = \frac{-3(2e^{2x} + 3x^2)}{(e^{2x} + x^3)^4}\)
| Questions | Answers |
|---|---|
| 1. \(f(x) = (2x + 5)^3\) | \(f'(x)=6(2x + 5)^2\) |
| 2. \(f(x) = \sqrt{3x^2 + 4}\) | \(f'(x)=\frac{3x}{\sqrt{3x^2 + 4}}\) |
| 3. \(f(x) = \frac{1}{(5x – 7)^2}\) | \(f'(x)=-\frac{10}{(5x – 7)^3}\) |
| 4. \(f(x) = (x^3 + 2x)^4\) | \(f'(x)=4(x^3 + 2x)^3(3x^2 + 2)\) |
| 5. \(f(x) = \sqrt{(x^2 + 1)^3}\) | \(f'(x)=3x\sqrt{x^2 + 1}\) |
| 6. \(f(x) = \frac{(4x – 1)^5}{2}\) | \(f'(x)=10(4x – 1)^4\) |
| 7. \(f(x) = (x^2 + 3x + 5)^3\) | \(f'(x)=3(x^2 + 3x + 5)^2(2x + 3)\) |
| 8. \(f(x) = \sqrt[4]{2x + 7}\) | \(f'(x)=\frac{1}{2(2x + 7)^{3/4}}\) |
| 9. \(f(x) = (x^2 + 4)^{5/3}\) | \(f'(x)=\frac{10x}{3}(x^2 + 4)^{2/3}\) |
| 10. \(f(x) = \frac{1}{(3x^2 – 2x + 1)^4}\) | \(f'(x)=-\frac{24x – 8}{(3x^2 – 2x + 1)^5}\) |
| 11. \(f(x) = \frac{7}{5 – 2x}\) | \(f'(x)=\frac{14}{(5 – 2x)^2}\) |
| 12. \(f(x) = \frac{1}{\sqrt{4 – x^2}}\) | \(f'(x)=\frac{x}{(4 – x^2)^{3/2}}\) |
| 13. \(f(x) = \left( \frac{1}{2}x + 3 \right)^7\) | \(f'(x)=\frac{7}{2}\left( \frac{1}{2}x + 3 \right)^6\) |
| 14. \(f(x) = \frac{1}{(x^3 – 4x + 1)^{2/3}}\) | \(f'(x)=-\frac{2(3x^2 – 4)}{3(x^3 – 4x + 1)^{5/3}}\) |
| 15. \(f(x) = (5x + 1)^{2/5}\) | \(f'(x)=\frac{2}{(5x + 1)^{3/5}}\) |