Questions on differentiation (Chain Rule)

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: $f'(x) = \frac{-3(2e^{2x} + 3x^2)}{(e^{2x} + x^3)^4}$

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. $f(x) = (5x + 1)^{2/5}$	$f'(x)=\frac{2}{(5x + 1)^{3/5}}$