# Ch05. Continuity and Differentiability (Part -3)

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The whole chapter Continuity and Differentiability is divided into three courses. This course is the part 3. The course is based on the assignments on Derivatives by Ashish Kumar (Agam Sir), which have questions from NCERT Exemplar, Board’s Question Bank, R. D. Sharma etc. The PDF of assignments can be downloaded within the course. Please note, all three courses based on Continuity and Differentiability do not have Rolle’s Theorem, Mean Value Theorem, NCERT Exercise 5.8 and questions based on them in Miscellaneous exercise.

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The following list of questions are just meant for reference before purchasing membership. The list might or might not include NCERT Questions as it depends on the chapter/course. Some chapters have NCERT questions combined in the Assignments and some chapters have separate NCERT questions and Assignments. For complete details, please check the index of the course in the "About Course".

### Assignment – 4

Find the derivative of following functions w.r.t. x:

1.         {\displaystyle y={{\sin }^{-1}}\left( \,2ax\sqrt{1-{{a}^{2}}{{x}^{2}}}\, \right)}

2.         {\displaystyle y={{\cos }^{-1}}\left( \frac{x-{{x}^{-1}}}{x+{{x}^{-1}}} \right)}

3.         {\displaystyle y={{\tan }^{-1}}\frac{4\sqrt{x}}{1-4x}}

4.         {\displaystyle y={{\cos }^{-1}}\left( \frac{a+b\cos x}{b+a\cos x} \right)}

5.         {\displaystyle {{\sin }^{-1}}\left( \frac{1}{\sqrt{x+1}} \right)}

6.           {\displaystyle \frac{{{8}^{x}}}{{{x}^{8}}}}

7.         {\displaystyle \log \left( x+\sqrt{{{x}^{2}}+a} \right)}

8.         {\displaystyle {{\log }_{3}}(3x+5)}

9.         {\displaystyle {{e}^{{{\log }_{2}}x}}}

10.       {\displaystyle {{e}^{6{{\log }_{e}}(x-1)}},\,\,x>1}

11.       {\displaystyle {{\sec }^{-1}}\sqrt{x}+\text{cose}{{\text{c}}^{-1}}\sqrt{x},\,\,\,x\ge 1}

12.       {\displaystyle {{\sin }^{-1}}({{x}^{{}^{7}/{}_{2}}})}

13.       {\displaystyle {{\log }_{x}}5}

14.       {\displaystyle {{2}^{{{\cos }^{2}}x}}}

15.       {\displaystyle \log \,[\log \,(\log {{x}^{5}})]}

### Assignment – 5

Differentiate the following functions w.r.t.  x

1. {\displaystyle x={{e}^{{{\tan }^{-1}}\left( \frac{y-{{x}^{2}}}{{{x}^{2}}} \right)}}}
2. {\displaystyle {{\sin }^{m}}x.{{\cos }^{n}}x}
3. {\displaystyle y={{x}^{\tan x}}+\sin {{x}^{\cos x}}}
4. {\displaystyle y=\frac{{{x}^{\frac{1}{2}}}{{(1-2x)}^{\frac{2}{3}}}}{{{(2-3x)}^{\frac{3}{4}}}{{(3-4x)}^{\frac{4}{5}}}}}
5. {\displaystyle y={{(1+{{x}^{-1}})}^{x}}}
6. {\displaystyle {{e}^{{{x}^{x}}}}}
7. {\displaystyle \log \tan \frac{x}{2}}
8. {\displaystyle {{10}^{\log \sin x}}}
9. {\displaystyle {{({{e}^{x}})}^{x}}}
10. {\displaystyle y={{(\tan x)}^{\cot x}}+{{(\cot x)}^{\tan x}}}
11. {\displaystyle y=\frac{{{(1-2x)}^{\frac{2}{3}}}{{(1+3x)}^{\frac{-3}{4}}}}{{{(1-6x)}^{\frac{5}{6}}}{{(1+7x)}^{\frac{-6}{7}}}}}
12. {\displaystyle y={{(\log x)}^{x}}+{{({{\sin }^{-1}}x)}^{\sin x}}}
13. {\displaystyle y=\frac{{{(1-x)}^{\frac{1}{2}}}{{(2-{{x}^{2}})}^{\frac{2}{3}}}}{{{(3-{{x}^{3}})}^{\frac{3}{4}}}{{(4-{{x}^{4}})}^{\frac{4}{5}}}}}
14. {\displaystyle y=\frac{{{x}^{3}}\sqrt{{{x}^{2}}+4}}{\sqrt{{{x}^{2}}+3}}}
15. {\displaystyle {{[\,1-{{x}^{2}}]}^{\frac{3}{2}}}.{{\sin }^{-1}}x}
16. {\displaystyle y=\frac{x\sqrt{{{x}^{2}}-4{{a}^{2}}}}{\sqrt{{{x}^{2}}-{{a}^{2}}}}}
17. {\displaystyle y={{\tan }^{-1}}\left( \frac{{{x}^{{}^{1}/{}_{3}}}+{{a}^{{}^{1}/{}_{3}}}}{1-{{a}^{{}^{1}/{}_{3}}}{{x}^{{}^{1}/{}_{3}}}} \right)}

### Assignment – 6

1. If {\displaystyle x={{e}^{\cos 2t}}} and {\displaystyle y={{e}^{\sin 2t}}} , prove that {\displaystyle \frac{dy}{dx}=\frac{-y\log x}{x\log y}} .
2. If {\displaystyle y={{({{\tan }^{-1}}x)}^{2}}} , then prove that {\displaystyle {{({{x}^{2}}+1)}^{2}}{{y}_{2}}+2x({{x}^{2}}+1){{y}_{1}}=2} .

Find {\displaystyle \frac{dy}{dx}} :

1. {\displaystyle x=t+\frac{1}{t},\,\,y=t-\frac{1}{t}}
2. {\displaystyle \begin{cases} & x={{e}^{\theta }}\left( \theta +\frac{1}{\theta } \right), \\  & y={{e}^{-\theta }}\left( \theta -\frac{1}{\theta } \right) \\ \end{cases}}
3. {\displaystyle \begin{cases} & x=3\cos \theta -2{{\cos }^{3}}\theta , \\  & y=3\sin \theta -2{{\sin }^{3}}\theta  \\ \end{cases}}
4. {\displaystyle \begin{cases} & x=\sin t\sqrt{\cos 2t}, \\  & y=\cos t\sqrt{\cos 2t} \\ \end{cases}}
5. {\displaystyle \begin{cases} & \sin x=\frac{2t}{1+{{t}^{2}}}, \\  & \tan y=\frac{2t}{1-{{t}^{2}}} \\ \end{cases}}
6. {\displaystyle x=\frac{1+\log t}{{{t}^{2}}},\,\,\,\,y=\frac{3+2\log t}{t}}
7. {\displaystyle y={{a}^{t+\frac{1}{t}}},\,\,\,\,x={{\left( t+\frac{1}{t} \right)}^{a}}} (NCERT Example – 47)
8. {\displaystyle {{x}^{\frac{2}{3}}}+{{y}^{\frac{2}{3}}}={{a}^{\frac{2}{3}}}} (NCERT Example – 37)
9. {\displaystyle {{x}^{y}}.{{y}^{x}}=1}
10. {\displaystyle {{y}^{\cot x}}+{{({{\tan }^{-1}}x)}^{y}}=1}
11. {\displaystyle y=x\cos y+y\cos x}
12. {\displaystyle \begin{cases} & y=12(1-\cos t), \\  & x=10(t-\sin t) \\ \end{cases}}
13. If {\displaystyle x=a(\cos t+t\sin t)} and {\displaystyle y=a(\sin t-t\cos t)} , find {\displaystyle \frac{{{d}^{2}}y}{d{{x}^{2}}}} .

### Assignment – 7

1. Differentiate {\displaystyle {{\sin }^{2}}x} w.r.t. {\displaystyle {{e}^{\cos x}}} . (NCERT Example – 48)
2. If {\displaystyle \sqrt{1-{{x}^{6}}}+\sqrt{1-{{y}^{6}}}={{a}^{3}}({{x}^{3}}-{{y}^{3}})} , prove that {\displaystyle \frac{dy}{dx}=\frac{{{x}^{2}}}{{{y}^{2}}}\sqrt{\frac{1-{{y}^{6}}}{1-{{x}^{6}}}}} .
3. If {\displaystyle x=a\sin 2t(1+\cos 2t)} and {\displaystyle y=b\cos 2t(1-\cos 2t)} , show that {\displaystyle {{\left( \frac{dy}{dx} \right)}_{t=\frac{\pi }{4}}}=\frac{b}{a}} .
4. If {\displaystyle x=3\sin t-\sin 3t} and {\displaystyle y=3\cos t-\cos 3t} , find {\displaystyle \frac{dy}{dx}} at {\displaystyle t=\frac{\pi }{3}} .
5. Differentiate {\displaystyle \frac{x}{\sin x}} w. r. t. {\displaystyle \sin x} .

Differentiate following w.r.t.  x

1. {\displaystyle \sin (xy)+\frac{x}{y}={{x}^{2}}-y}
2. {\displaystyle {{({{x}^{2}}+{{y}^{2}})}^{2}}=xy}
3. {\displaystyle \sec (x+y)=xy}
4. If {\displaystyle \frac{x}{x-y}=\log \frac{a}{x-y}} , prove that {\displaystyle \frac{dy}{dx}=2-\frac{x}{y}} .
5. Differentiate {\displaystyle {{x}^{\sin x}}} with respect to {\displaystyle {{\left( \sin x \right)}^{x}}} .
6. If {\displaystyle f(x)=\left| \,\begin{matrix} x & {{x}^{2}} & {{x}^{3}}  \\   1 & 2x & 3{{x}^{2}}  \\   0 & 2 & 6x  \\\end{matrix}\, \right|} , find {\displaystyle f\,'(x)} .
7. Differentiate {\displaystyle {{\tan }^{-1}}\left( \frac{\sqrt{1+{{x}^{2}}}-1}{x} \right) {\displaystyle w. r. t. } </span>{{\tan }^{-1}}x} when {\displaystyle x\ne 0} .
8. Differentiate {\displaystyle {{\tan }^{-1}}\left[ \frac{\sqrt{1+{{x}^{2}}}-\sqrt{1-{{x}^{2}}}}{\sqrt{1+{{x}^{2}}}+\sqrt{1-{{x}^{2}}}} \right]} with respect to {\displaystyle {{\cos }^{-1}}({{x}^{2}})} .
9. Differentiate {\displaystyle {{\left( \log x \right)}^{\tan x}}} with respect to {\displaystyle \sin (m{{\cos }^{-1}}x)} .
10. If {\displaystyle a{{x}^{2}}+2hxy+b{{y}^{2}}+2gx+2fy+c=0} , then show that {\displaystyle \frac{dy}{dx}.\frac{dx}{dy}=1} .

### Assignment – 8

1. If {\displaystyle x={{e}^{\frac{x}{y}}}} , prove that {\displaystyle \frac{dy}{dx}=\frac{x-y}{x\log x}} .
2. If {\displaystyle {{y}^{x}}={{e}^{y-x}}} , prove that {\displaystyle \frac{dy}{dx}=\frac{{{(1+\log y)}^{2}}}{\log y}} .
3. If {\displaystyle y={{(\cos x)}^{{{(\cos x)}^{(\cos x)...\infty }}}}} , show that {\displaystyle \frac{dy}{dx}=\frac{{{y}^{2}}\tan x}{y\log \cos x-1}} .
4. If {\displaystyle y={{\tan }^{-1}}x} , find {\displaystyle \frac{{{d}^{2}}y}{d{{x}^{2}}}} in terms of {\displaystyle y} alone.
5. If {\displaystyle {{x}^{m}}.{{y}^{n}}={{(x+y)}^{m+n}}} , prove that {\displaystyle \frac{dy}{dx}=\frac{y}{x}} and {\displaystyle \frac{{{d}^{2}}y}{d{{x}^{2}}}=0} .
6. If {\displaystyle x=\sin t} and {\displaystyle y=\sin pt} , prove that {\displaystyle (1-{{x}^{2}})\frac{{{d}^{2}}y}{d{{x}^{2}}}-x\frac{dy}{dx}+{{p}^{2}}y=0} .
7. Differentiate w.r.t. {\displaystyle x} : {\displaystyle y={{x}^{\tan x}}+\sqrt{\frac{{{x}^{2}}+1}{2}}}
8. If {\displaystyle {{e}^{x}}+{{e}^{y}}={{e}^{x+y}}} , prove that {\displaystyle \frac{dy}{dx}=-{{e}^{y-x}}} .
9. If {\displaystyle y={{\sin }^{-1}}x\sqrt{1-x}-\sqrt{x}\sqrt{1-{{x}^{2}}}} , find {\displaystyle \frac{dy}{dx}} .
10. Differentiate {\displaystyle {{\sin }^{-1}}\left[ \frac{3x+4\sqrt{1-{{x}^{2}}}}{5} \right]} w.r.t. x.
11. Differentiate {\displaystyle {{\sin }^{-1}}\left( \frac{{{2}^{x+1}}{{.3}^{x}}}{1+{{(36)}^{x}}} \right)} w.r.t. x.
12. Differentiate {\displaystyle {{\tan }^{-1}}\left( \frac{\sqrt{1-{{x}^{2}}}}{x} \right)} w.r.t. {\displaystyle {{\cos }^{-1}}\left( 2x\sqrt{1-{{x}^{2}}} \right)} , where {\displaystyle x\ne 0} .
13. If {\displaystyle y={{x}^{{{x}^{x}}}}} , then find {\displaystyle \frac{dy}{dx}} .
14. If {\displaystyle f(x)=x+7} and {\displaystyle g(x)=x-7} , {\displaystyle x\in \mathbf{R}} , then find {\displaystyle \frac{d}{dx}(fog)(x)} .
15. Differentiate {\displaystyle {{\sin }^{2}}({{\theta }^{2}}+1)} w.r.t. {\displaystyle {{\theta }^{2}}} .
16. If {\displaystyle f(x)={{x}^{2}}g(x)} and {\displaystyle g(1)=6,\,\,g'(x)=3} , find the value of {\displaystyle f'(1)} .
17. Find {\displaystyle \frac{dy}{dx}} if {\displaystyle y={{\sin }^{-1}}\left( \frac{\sqrt{x}-1}{\sqrt{x}+1} \right)+{{\sec }^{-1}}\left( \frac{\sqrt{x}+1}{\sqrt{x}-1} \right)} .
18. If {\displaystyle y=x\log \left( \frac{x}{a+bx} \right)} prove that {\displaystyle {{x}^{3}}\frac{{{d}^{2}}y}{d{{x}^{2}}}={{\left( x\frac{dy}{dx}-y \right)}^{2}}} .
19. If {\displaystyle f(x)=\sqrt{{{x}^{2}}+1},\,\,g(x)=\frac{x+1}{{{x}^{2}}+1}} and {\displaystyle h(x)=2x-3} , find {\displaystyle f'[h'(g'(x))]} .
20. If {\displaystyle {{y}^{\frac{1}{m}}}+{{y}^{-\frac{1}{m}}}=2x} then prove that {\displaystyle ({{x}^{2}}-1){{y}_{2}}+x{{y}_{1}}={{m}^{2}}y} .
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