Questions on differentiation (Chain Rule)

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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:

  1. Outer function: e^u, where u = 4x^3 - 2x.
  2. Differentiate outer: \frac{d}{du} e^u = e^u.
  3. Differentiate inner: \frac{du}{dx} = 12x^2 - 2.
  4. Apply chain rule: f'(x) = e^u \cdot \frac{du}{dx}.
  5. 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:

  1. Outer function: u^4, where u = \ln(5x + 3).
  2. Differentiate outer: 4u^3.
  3. Differentiate inner: \frac{du}{dx} = \frac{5}{5x + 3}.
  4. Apply chain rule: f'(x) = 4u^3 \cdot \frac{du}{dx}.
  5. 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:

  1. Rewrite: f(x) = (3x^2 + 2x - 1)^{-5}.
  2. Outer function: u^{-5}, where u = 3x^2 + 2x - 1.
  3. Differentiate outer: -5u^{-6}.
  4. Differentiate inner: \frac{du}{dx} = 6x + 2.
  5. 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:

  1. Rewrite: f(x) = (\ln(2x + 1))^{1/2}.
  2. Outer function: u^{1/2}, where u = \ln(2x + 1).
  3. Differentiate outer: \frac{1}{2}u^{-1/2}.
  4. Differentiate inner: \frac{du}{dx} = \frac{2}{2x + 1}.
  5. 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:

  1. Outer function: u^{-3}, where u = e^{2x} + x^3.
  2. Differentiate outer: -3u^{-4}.
  3. Differentiate inner: \frac{du}{dx} = 2e^{2x} + 3x^2.
  4. Apply chain rule: f'(x) = -3(e^{2x} + x^3)^{-4} \cdot (2e^{2x} + 3x^2)
  5. Simplify: f'(x) = \frac{-3(2e^{2x} + 3x^2)}{(e^{2x} + x^3)^4}
QuestionsAnswers
1. f(x) = (2x + 5)^3f'(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)^4f'(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)^3f'(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)^7f'(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}}

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