Examples Differentiation by Chain Rule

Practice Questions (with Step-by-Step Solutions)

Question 1.

Differentiate [f(x)=(5x+3)^{4}].

Step-by-Step Solution:

1. Identify inner and outer: inner [g(x)=5x+3], outer [F(u)=u^{4}].
2. [\dfrac{d}{du}(u^{4})=4u^{3}].
3. [\dfrac{d}{dx}(5x+3)=5].
4. Apply chain rule:
   [\dfrac{d}{dx}\big((5x+3)^{4}\big)][=4(5x+3)^{3}\cdot 5].
5. Simplify:
   [=20(5x+3)^{3}].

Answer:
[\dfrac{d}{dx}\big((5x+3)^{4}\big)][=20(5x+3)^{3}].

Question 2.

Differentiate [f(x)=\sin(3x^{2})].

Step-by-Step Solution:

1. Inner: [g(x)=3x^{2}], outer: [F(u)=\sin u].
2. [\dfrac{d}{du}(\sin u)=\cos u].
3. [\dfrac{d}{dx}(3x^{2})=6x].
4. Chain rule:
   [\dfrac{d}{dx}\big(\sin(3x^{2})\big)][=\cos(3x^{2})\cdot 6x].
5. Final:
   [=6x\cos(3x^{2})].

Answer:
[\dfrac{d}{dx}\big(\sin(3x^{2})\big)][=6x\cos(3x^{2})].

Question 3.

Differentiate [f(x)=e^{\sin x}].

Step-by-Step Solution:

1. Inner: [g(x)=\sin x], outer: [F(u)=e^{u}].
2. [\dfrac{d}{du}(e^{u})=e^{u}].
3. [\dfrac{d}{dx}(\sin x)=\cos x].
4. Chain rule:
   [\dfrac{d}{dx}\big(e^{\sin x}\big)][=e^{\sin x}\cdot \cos x].
5. Final tidy form:
   [= \cos x e^{\sin x}].

Answer:
[\dfrac{d}{dx}\big(e^{\sin x}\big)][=\cos x; e^{\sin x}].

Question 4.

Differentiate [f(x)=(\ln x)^{3}], domain [x>0].

Step-by-Step Solution:

1. Inner: [g(x)=\ln x], outer: [F(u)=u^{3}].
2. [\dfrac{d}{du}(u^{3})=3u^{2}].
3. [\dfrac{d}{dx}(\ln x)=\dfrac{1}{x}].
4. Chain rule:
   [\dfrac{d}{dx}\big((\ln x)^{3}\big)][=3(\ln x)^{2}\cdot \dfrac{1}{x}].
5. Final:
   [= \dfrac{3(\ln x)^{2}}{x}].

Answer:
[\dfrac{d}{dx}\big((\ln x)^{3}\big)][=\dfrac{3(\ln x)^{2}}{x},\quad x>0.]

Question 5.

Differentiate [f(x)=\sqrt{1+x^{4}}][=(1+x^{4})^{1/2}].

Step-by-Step Solution:

1. Inner: [g(x)=1+x^{4}], outer: [F(u)=u^{1/2}].
2. [\dfrac{d}{du}(u^{1/2})=\dfrac{1}{2}u^{-1/2}].
3. [\dfrac{d}{dx}(1+x^{4})=4x^{3}].
4. Chain rule:
   [\dfrac{d}{dx}\big((1+x^{4})^{1/2}\big)][=\dfrac{1}{2}(1+x^{4})^{-1/2}\cdot 4x^{3}].
5. Simplify constants:
   [=2x^{3}(1+x^{4})^{-1/2}] → or [\dfrac{2x^{3}}{\sqrt{1+x^{4}}}].

Answer:
[\dfrac{d}{dx}\big(\sqrt{1+x^{4}}\big)=\dfrac{2x^{3}}{\sqrt{1+x^{4}}}].

Question 6.

Differentiate [f(x)=\tan^{-1}(2x+1)].

Step-by-Step Solution:

1. Inner: [g(x)=2x+1], outer: [F(u)=\tan^{-1}u].
2. [\dfrac{d}{du}(\tan^{-1}u)][=\dfrac{1}{1+u^{2}}].
3. [\dfrac{d}{dx}(2x+1)=2].
4. Chain rule:
   [\dfrac{d}{dx}\big(\tan^{-1}(2x+1)\big)][=\dfrac{1}{1+(2x+1)^{2}}\cdot 2].
5. Final:
   [= \dfrac{2}{1+(2x+1)^{2}}].

Answer:
[\dfrac{d}{dx}\big(\tan^{-1}(2x+1)\big)][=\dfrac{2}{1+(2x+1)^{2}}].

Question 7.

Differentiate [f(x)=\sec(x^{2})].

Step-by-Step Solution:

1. Inner: [g(x)=x^{2}], outer: [F(u)=\sec u].
2. [\dfrac{d}{du}(\sec u)=\sec u \tan u].
3. [\dfrac{d}{dx}(x^{2})=2x].
4. Chain rule:
   [\dfrac{d}{dx}\big(\sec(x^{2})\big)][=\sec(x^{2})\tan(x^{2})\cdot 2x].
5. Final tidy form:
   [=2x\sec(x^{2})\tan(x^{2})].

Answer:
[\dfrac{d}{dx}\big(\sec(x^{2})\big)][=2x\sec(x^{2})\tan(x^{2})].

Question 8.

Differentiate [f(x)=(3x^{2}+1)^{1/3}].

Step-by-Step Solution:

1. Inner: [g(x)=3x^{2}+1], outer: [F(u)=u^{1/3}].
2. [\dfrac{d}{du}(u^{1/3})=\dfrac{1}{3}u^{-2/3}].
3. [\dfrac{d}{dx}(3x^{2}+1)=6x].
4. Chain rule:
   [\dfrac{d}{dx}\big((3x^{2}+1)^{1/3}\big)][=\dfrac{1}{3}(3x^{2}+1)^{-2/3}\cdot 6x].
5. Simplify constants:
   [=2x(3x^{2}+1)^{-2/3}] → or [\dfrac{2x}{(3x^{2}+1)^{2/3}}].

Answer:
[\dfrac{d}{dx}\big((3x^{2}+1)^{1/3}\big)][=\dfrac{2x}{(3x^{2}+1)^{2/3}}].

Question 9.

Differentiate [f(x)=\sin^{-1}!\big(\dfrac{x}{2}\big)], domain [|x|<2].

Step-by-Step Solution:

1. Inner: [g(x)=\dfrac{x}{2}], outer: [F(u)=\sin^{-1}u].
2. [\dfrac{d}{du}(\sin^{-1}u)][=\dfrac{1}{\sqrt{1-u^{2}}}].
3. [\dfrac{d}{dx}\big(\dfrac{x}{2}\big)][=\dfrac{1}{2}].
4. Chain rule:
   [\dfrac{d}{dx}\Big(\sin^{-1}\big(\dfrac{x}{2}\big)\Big)][=\dfrac{1}{\sqrt{1-(\tfrac{x}{2})^{2}}}\cdot \dfrac{1}{2}].
5. Simplify inside radical: [1-(\dfrac{x}{2})^{2}][=1-\dfrac{x^{2}}{4}][=\dfrac{4-x^{2}}{4}]. So sqrt = [\dfrac{\sqrt{4-x^{2}}}{2}]. Thus expression becomes:
   [\dfrac{1}{(\sqrt{4-x^{2}}/2)}\cdot \dfrac{1}{2}][ = \dfrac{2}{\sqrt{4-x^{2}}}\cdot \dfrac{1}{2}][ = \dfrac{1}{\sqrt{4-x^{2}}}].
6. Final neat form:
   [= \dfrac{1}{\sqrt{4-x^{2}}}], valid for [|x|<2].

Answer:
[\dfrac{d}{dx}\Big(\sin^{-1}\big(\dfrac{x}{2}\big)\Big)][=\dfrac{1}{\sqrt{4-x^{2}}},\quad |x|<2.]

Question 10.

Differentiate [f(x)=(e^{x}+x)^{2}].

Step-by-Step Solution:

1. Inner: [g(x)=e^{x}+x], outer: [F(u)=u^{2}].
2. [\dfrac{d}{du}(u^{2})=2u].
3. [\dfrac{d}{dx}(e^{x}+x)=e^{x}+1] (differentiate termwise).
4. Chain rule:
   [\dfrac{d}{dx}\big((e^{x}+x)^{2}\big)][=2(e^{x}+x)\cdot (e^{x}+1)].
5. Optionally expand (not necessary): final compact form is preferable.

Answer:
[\dfrac{d}{dx}\big((e^{x}+x)^{2}\big)][=2(e^{x}+x)(e^{x}+1)].