Continuity and Differentiability Part - 3

Questions Discussed:

Assignment - 4

Find the derivative of following functions w.r.t. { \displaystyle x } :

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Questions Discussed:

Assignment - 5

Differentiate the following functions w.r.t. { \displaystyle x }

Question 1. { \displaystyle x={{e}^{{{\tan }^{-1}}\left( \frac{y-{{x}^{2}}}{{{x}^{2}}} \right)}} }

Question 2. { \displaystyle {{\sin }^{m}}x.{{\cos }^{n}}x }

Question 3. { \displaystyle y={{x}^{\tan x}}+\sin {{x}^{\cos x}} }

Question 4. { \displaystyle y=\frac{{{x}^{\frac{1}{2}}}{{(1-2x)}^{\frac{2}{3}}}}{{{(2-3x)}^{\frac{3}{4}}}{{(3-4x)}^{\frac{4}{5}}}} }

Question 5. { \displaystyle y={{(1+{{x}^{-1}})}^{x}} }

Question 6. { \displaystyle {{e}^{{{x}^{x}}}} }

Question 7. { \displaystyle \log \tan \frac{x}{2} }

Question 8. { \displaystyle {{10}^{\log \sin x}} }

Question 9. { \displaystyle {{({{e}^{x}})}^{x}} }

Questions Discussed:

Assignment - 5

Question 10. { \displaystyle y={{(\tan x)}^{\cot x}}+{{(\cot x)}^{\tan x}} }

Question 11. { \displaystyle y=\frac{{{(1-2x)}^{\frac{2}{3}}}{{(1+3x)}^{\frac{-3}{4}}}}{{{(1-6x)}^{\frac{5}{6}}}{{(1+7x)}^{\frac{-6}{7}}}} }

Question 12. { \displaystyle y={{(\log x)}^{x}}+{{({{\sin }^{-1}}x)}^{\sin x}} }

Question 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}}}} }

Question 14. { \displaystyle y=\frac{{{x}^{3}}\sqrt{{{x}^{2}}+4}}{\sqrt{{{x}^{2}}+3}} }

Question 15. { \displaystyle {{[\,1-{{x}^{2}}]}^{\frac{3}{2}}}.{{\sin }^{-1}}x }

Question 16. { \displaystyle y=\frac{x\sqrt{{{x}^{2}}-4{{a}^{2}}}}{\sqrt{{{x}^{2}}-{{a}^{2}}}} }

Question 17. { \displaystyle y={{\tan }^{-1}}\left( \frac{{{x}^{{}^{1}/{}_{3}}}+{{a}^{{}^{1}/{}_{3}}}}{1-{{a}^{{}^{1}/{}_{3}}}{{x}^{{}^{1}/{}_{3}}}} \right) }

Questions Discussed:

Assignment - 6

Question 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} } .

Question 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} } :

Question 3. { \displaystyle x=t+\frac{1}{t},\,\,y=t-\frac{1}{t} }

Question 4. { \displaystyle x={{e}^{\theta }}\left( \theta +\frac{1}{\theta } \right) } , { \displaystyle y={{e}^{-\theta }}\left( \theta -\frac{1}{\theta } \right) }

Question 5. { \displaystyle x=3\cos \theta -2{{\cos }^{3}}\theta } , { \displaystyle y=3\sin \theta -2{{\sin }^{3}}\theta }

Question 6. { \displaystyle x=\sin t\sqrt{\cos 2t} } , { \displaystyle y=\cos t\sqrt{\cos 2t} }

Question 7. { \displaystyle \sin x=\frac{2t}{1+{{t}^{2}}} } , { \displaystyle \tan y=\frac{2t}{1-{{t}^{2}}} }

Question 8. { \displaystyle x=\frac{1+\log t}{{{t}^{2}}},\,y=\frac{3+2\log t}{t} }

Questions Discussed:

Assignment - 6

Question 9. { \displaystyle y={{a}^{t+\frac{1}{t}}},\,\,\,\,x={{\left( t+\frac{1}{t} \right)}^{a}} } (NCERT Example – 47)

Question 10. { \displaystyle {{x}^{\frac{2}{3}}}+{{y}^{\frac{2}{3}}}={{a}^{\frac{2}{3}}} } (NCERT Example – 37)

Question 11. { \displaystyle {{x}^{y}}.{{y}^{x}}=1 }

Question 12. { \displaystyle {{y}^{\cot x}}+{{({{\tan }^{-1}}x)}^{y}}=1 }

Question 13. { \displaystyle y=x\cos y+y\cos x }

Question 14. { \displaystyle y=12(1-\cos t)} , { \displaystyle x=10(t-\sin t) }

Question 15. 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}}} } .

Questions Discussed:

Assignment - 7

Question 1. Differentiate { \displaystyle {{\sin }^{2}}x } w.r.t. { \displaystyle {{e}^{\cos x}} } . (NCERT Example – 48)

Question 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}}}} } .

Question 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} } .

Question 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} } .

Question 5. Differentiate { \displaystyle \frac{x}{\sin x} } w. r. t. { \displaystyle \sin x } .

Differentiate following w.r.t. { \displaystyle x }

Question 6. { \displaystyle \sin (xy)+\frac{x}{y}={{x}^{2}}-y }

Question 7. { \displaystyle {{({{x}^{2}}+{{y}^{2}})}^{2}}=xy }

Question 8. { \displaystyle \sec (x+y)=xy }

Questions Discussed:

Assignment - 7

Question 9. If { \displaystyle \frac{x}{x-y}=\log \frac{a}{x-y} } , prove that { \displaystyle \frac{dy}{dx}=2-\frac{x}{y} } .

Question 10. Differentiate { \displaystyle {{x}^{\sin x}} } with respect to { \displaystyle {{\left( \sin x \right)}^{x}} } .

Question 11. 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) } .

Question 12. Differentiate { \displaystyle {{\tan }^{-1}}\left( \frac{\sqrt{1+{{x}^{2}}}-1}{x} \right) { \displaystyle w. r. t. } {{\tan }^{-1}}x } when { \displaystyle x\ne 0 } .

Question 13. 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}}) } .

Question 14. Differentiate { \displaystyle {{\left( \log x \right)}^{\tan x}} } with respect to { \displaystyle \sin (m{{\cos }^{-1}}x) } .

Question 15. If { \displaystyle a{{x}^{2}}+2hxy+b{{y}^{2}}+2gx+2fy+c=0 } , then show that { \displaystyle \frac{dy}{dx}.\frac{dx}{dy}=1 } .

Questions Discussed:

Assignment - 8

Question 1. If { \displaystyle x={{e}^{\frac{x}{y}}} } , prove that { \displaystyle \frac{dy}{dx}=\frac{x-y}{x\log x} } .

Question 2. If { \displaystyle {{y}^{x}}={{e}^{y-x}} } , prove that { \displaystyle \frac{dy}{dx}=\frac{{{(1+\log y)}^{2}}}{\log y} } .

Question 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} } .

Question 4. If { \displaystyle y={{\tan }^{-1}}x } , find { \displaystyle \frac{{{d}^{2}}y}{d{{x}^{2}}} } in terms of { \displaystyle y } alone.

Question 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 } .

Question 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 } .

Question 7. Differentiate w.r.t. { \displaystyle x } : { \displaystyle y={{x}^{\tan x}}+\sqrt{\frac{{{x}^{2}}+1}{2}} }

Question 8. If { \displaystyle {{e}^{x}}+{{e}^{y}}={{e}^{x+y}} } , prove that { \displaystyle \frac{dy}{dx}=-{{e}^{y-x}} } .

Question 9. If { \displaystyle y={{\sin }^{-1}}x\sqrt{1-x}-\sqrt{x}\sqrt{1-{{x}^{2}}} } , find { \displaystyle \frac{dy}{dx} } .

Question 10. Differentiate { \displaystyle {{\sin }^{-1}}\left[ \frac{3x+4\sqrt{1-{{x}^{2}}}}{5} \right] } w.r.t. x.

Questions Discussed:

Assignment - 8

Question 11. Differentiate { \displaystyle {{\sin }^{-1}}\left( \frac{{{2}^{x+1}}{{.3}^{x}}}{1+{{(36)}^{x}}} \right) } w.r.t. x.

Question 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 } .

Question 13. If { \displaystyle y={{x}^{{{x}^{x}}}} } , then find { \displaystyle \frac{dy}{dx} } .

Question 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) } .

Question 15. Differentiate { \displaystyle {{\sin }^{2}}({{\theta }^{2}}+1) } w.r.t. { \displaystyle {{\theta }^{2}} } .

Question 16. If { \displaystyle f(x)={{x}^{2}}g(x) } and { \displaystyle g(1)=6,\,\,g'(x)=3 } , find the value of { \displaystyle f'(1) } .

Question 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) } .

Question 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}} } .

Question 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))] } .

Question 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 } .