Continuity and Differentiability Part - 2

Questions Discussed:

NCERT Exercise 5.2

Differentiate the functions with respect to x in Exercise 1 to 8

Question 1. { \displaystyle \sin(x^2 + 5) }

Question 2. { \displaystyle \cos (\sin x) }

Question 3. { \displaystyle \sin (ax + b) }

Question 4. { \displaystyle \sec (\tan \sqrt{x}) }

Question 5. { \displaystyle \frac{\sin(ax + b)}{\cos(cx + d)} }

Question 6. { \displaystyle \cos x^3 . \sin^2(x^5) }

Question 7. { \displaystyle 2\sqrt{cot(x^2)} }

Question 8. { \displaystyle \cos(\sqrt{x}) }

NCERT Examples

Example 26: Find the derivative of f given by { \displaystyle f(x) = \sin^{-1}x } assuming it exists.

Example 27: Find the derivative of f given by { \displaystyle f(x) = \tan^{-1}x } assuming it exists.

NCERT Exercise 5.3

Find { \displaystyle \frac{dy}{dx} } in the following:

Question 1: { \displaystyle 2x + 3y = \sin x }

Question 2: { \displaystyle 2x + 3y = \sin y }

Question 3: { \displaystyle ax + by^2 = \cos y }

Question 4: { \displaystyle xy + y^2 = tanx + y }

Question 5: { \displaystyle x^2 + xy + y^2 = 100 }

Question 6: { \displaystyle x^3 + x^2y + xy^2 + y^3 = 81 }

Question 7: { \displaystyle \sin^2 y + \cos xy = \kappa }

Question 8: { \displaystyle \sin^2 x + \cos^2 y = 1 }

Question 9: { \displaystyle y = \sin^{-1} \left ( \frac{2x}{1 + x^2} \right ) }

Question 10: { \displaystyle y = \tan^{-1} \left ( \frac{3x - x^3}{1 - 3x^2} \right ) } , { \displaystyle -\frac{1}{\sqrt{3}} < x < \frac{1}{\sqrt{3}} }

Question 11: { \displaystyle y = \cos^{-1} \left ( \frac{1 - x^2}{1 + x^2} \right ) } , { \displaystyle 0 < x < 1 }

Question 12: { \displaystyle y = \sin^{-1} \left ( \frac{1 - x^2}{1 + x^2} \right ) } , { \displaystyle 0 < x < 1 }

Question 13: { \displaystyle y = \cos^{-1} \left ( \frac{2x}{1 + x^2} \right ) } , { \displaystyle -1 < x < 1 }

Question 14: { \displaystyle y = \sin^{-1} \left ( 2x \sqrt{1-x^2} \right ) } , { \displaystyle -\frac{1}{\sqrt{2}} < x < \frac{1}{\sqrt{2}} }

Question 15: { \displaystyle y = \sec^{-1} \left ( \frac{1}{2x^2 - 1} \right ) } , { \displaystyle 0 < x < \frac{1}{\sqrt{2}} }

Questions Discussed:

NCERT Exercise 5.4

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

Question 1. { \displaystyle \frac{e^x}{\sin x} }

Question 2. { \displaystyle e^{\sin^{-1}x} }

Question 3. { \displaystyle e^{x^3} }

Question 4. { \displaystyle \sin (\tan^{-1} e^{-x}) }

Question 5. { \displaystyle \log (\cos e^x) }

Question 6. { \displaystyle e^x + e^{x^2} + ... + e^{x^5} }

Question 7. { \displaystyle \sqrt{e^{\sqrt{x}}}, x > 0 }

Question 8. { \displaystyle \log (\log x), x > 1 }

Question 9. { \displaystyle \frac{\cos x}{\log x}, x > 0 }

Question 10. { \displaystyle \cos(\log x + e^x), x > 0 }

NCERT Exercise 5.5

Differentiate the functions given in Exercises 1 to 11 w.r.t. { \displaystyle x }

Question 1. { \displaystyle \cos x. \cos 2x. \cos 3x }

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

Question 3. { \displaystyle (\log x)^{\cos x} }

Question 4. { \displaystyle x^x - 2^{\sin x} }

Question 5. { \displaystyle (x+3)^2 . (x+4)^3 . (x+5)^4 }

Question 6. { \displaystyle \left ( x + \frac{1}{x} \right )^x + x^{\left ( 1 + \frac{1}{x} \right )} }

Question 7. { \displaystyle (\log x)^x + x^{\log x} }

Question 8. { \displaystyle (\sin x)^x + \sin^{-1} \sqrt{x} }

Question 9. { \displaystyle x^{\sin x} + (\sin x)^{\cos x} }

Question 10. { \displaystyle x^{x \cos x} + \frac{x^2+1}{x^2-1} }

Question 11. { \displaystyle (x \cos x)^x + (x \sin x)^{\frac{1}{x}} }

Questions Discussed:

NCERT Exercise 5.5

Find { \displaystyle \frac{dy}{dx} } of the functions given in Exercises 12 to 15

Question 12. { \displaystyle x^y + y^x = 1 }

Question 13. { \displaystyle y^x = x^y }

Question 14. { \displaystyle (\cos x)^y = (\cos y)^x }

Question 15. { \displaystyle xy = e^{(x-y)} }

Question 16. Find the derivative of the function given by { \displaystyle f(x)=(1+x)(1+x^2) } { \displaystyle (1+x^4)(1+x^8) } and hence find { \displaystyle f'(1) }

Question 17. Differentiate { \displaystyle (x^2 - 5x + 8)(x^3 + 7x + 9) } in three ways mentioned below:
(i) by using product rule
(ii) by expanding the product to obtain a single polynomial.
(iii) by logarithmic differentiation.
Do they all give the same answer?

Question 18. If { \displaystyle u, v } and { \displaystyle w } are functions of { \displaystyle x } , then show that
{ \displaystyle \frac{d}{dx} (u.v.w) = \frac{du}{dx} v.w } + { \displaystyle u . \frac{dv}{dx} . w + u . v \frac{dw}{dx} }
in two ways - first by repeated application of product rule, second by logarithmic differentiation.

NCERT Exercise 5.6

If { \displaystyle x } and { \displaystyle y } are connected parametrically by the equations given in Exercises 1 to 10, without eliminating the parameter, Find { \displaystyle \frac{dy}{dx} } .

Question 1. { \displaystyle x = 2at^2, y = at^4 }

Question 2. { \displaystyle x = a\cos \theta, y = b \cos \theta }

Question 3. { \displaystyle x = \sin t, y = \cos 2t }

Question 4. { \displaystyle x = 4t, y = \frac{4}{t} }

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

Question 6. { \displaystyle x = a (\theta - \sin \theta) } , { \displaystyle y = a(1 + \cos \theta) }

Question 7. { \displaystyle x = \frac{\sin^{3}t}{\sqrt{\cos 2t}} } , { \displaystyle y = \frac{\cos^{3}t}{\sqrt{\cos 2t}} }

Question 8. { \displaystyle x = a \left ( \cos t + \log \tan \frac{t}{2} \right ) } , { \displaystyle y = a \sin t }

Question 9. { \displaystyle x = a \sec \theta, y = b \tan \theta }

Question 10. { \displaystyle x = a(\cos \theta + \theta \sin \theta) } , { \displaystyle y = a(\sin \theta - \theta \cos \theta) }

Question 11. If { \displaystyle x = \sqrt{a^{\sin^{-1}t}} } , { \displaystyle y = \sqrt{a^{\cos^{-1}t}} } , show that { \displaystyle \frac{dy}{dx} = -\frac{y}{x} }

Questions Discussed:

NCERT Exercise 5.7

Find the second order derivatives of the functions given in Exercises 1 to 10.

Question 1. { \displaystyle x^2 + 3x + 2 }

Question 2. { \displaystyle x^{20} }

Question 3. { \displaystyle x.\cos x }

Question 4. { \displaystyle \log x }

Question 5. { \displaystyle x^3 \log x }

Question 6. { \displaystyle e^x \sin 5x }

Question 7. { \displaystyle e^{6x} \cos 3x }

Question 8. { \displaystyle \tan^{-1}x }

Question 9. { \displaystyle \log(\log x) }

Question 10. { \displaystyle \sin (\log x) }

Question 11. If { \displaystyle y = 5 \cos x - 3 \sin x } , prove that { \displaystyle \frac{d^{2}y}{dx^2} + y = 0 } .

Question 12. If { \displaystyle y = \cos^{-1}x } , find { \displaystyle \frac{d^2y}{dx^2} } in terms of { \displaystyle y } alone.

Question 13. If { \displaystyle y = 3 \cos (\log x) + 4 \sin (\log x) } , show that { \displaystyle x^2y_2 + xy_1 + y = 0 } .

Question 14. If { \displaystyle y = Ae^{mx} + Be^{nx} } , show that { \displaystyle \frac{d^2y}{dx^2} - (m + n)\frac{dy}{dx} + mny = 0 } .

Question 15. If { \displaystyle y = 500e^{7x} + 600e^{-7x} } , show that { \displaystyle \frac{d^2y}{dx^2} = 49y } .

Question 16. If { \displaystyle e^y(x + 1) = 1 } , show that { \displaystyle \frac{d^2y}{dx^2} = \left ( \frac{dy}{dx} \right )^2 } .

Question 17. If { \displaystyle y = (\tan^{-1}x)^2 } , show that { \displaystyle (x^2 + 1)^2y_2 + 2x(x^2 + 1)y_1 = 2 } .

Questions Discussed:

NCERT Miscellaneous Exercise

Differentiate w.r.t. { \displaystyle x } the function in Exercises 1 to 11.

Question 1. { \displaystyle (3x^2 - 9x + 5)^9 }

Question 2. { \displaystyle \sin^3x + \cos^6x }

Question 3. { \displaystyle (5x)^{3\cos2x} }

Question 4. { \displaystyle \sin^{-1}(x\sqrt{x}), 0 \le x \le 1 }

Question 5. { \displaystyle \frac{\cos^{-1}\frac{x}{2}}{\sqrt{2x + 7}}, -2 < x < 2 }

Question 6. { \displaystyle \cot^{-1}\left [ \frac{\sqrt{1 + \sin x} + \sqrt{1 - \sin x}}{\sqrt{1 + \sin x} - \sqrt{1 - \sin x}} \right ] } , { \displaystyle 0 < x < \frac{\pi}{2} }

Question 7. { \displaystyle (\log x)^{\log x}, x > 1 }

Question 8. { \displaystyle \cos(a \cos x + b \sin x) } , for some constant { \displaystyle a } and { \displaystyle b } .

Question 9. { \displaystyle (\sin x - \cos x)^{(\sin x - \cos x)} } , { \displaystyle \frac{\pi}{4} < x < \frac{3\pi}{4} }

Question 10. { \displaystyle x^x + x^a + a^x + a^a } , for some fixed { \displaystyle a > 0 } and { \displaystyle x > 0 }

Question 11. { \displaystyle x^{x^2 - 3} + (x - 3)^{x^2} } , for { \displaystyle x > 3 }

Questions Discussed:

NCERT Miscellaneous Exercise

Question 12. Find { \displaystyle \frac{dy}{dx} } , if { \displaystyle y = 12 (1 - \cos t) } , { \displaystyle x = 10 (t- \sin t) } , { \displaystyle -\frac{\pi}{2} < t < \frac{\pi}{2} }

Question 13. Find { \displaystyle \frac{dy}{dx} } , if { \displaystyle y = \sin^{-1}x + \sin^{-1}\sqrt{1 - x^2} } , { \displaystyle 0 < x < 1 }

Question 14. If { \displaystyle x\sqrt{1 + y} + y\sqrt{1 + x} = 0 } , for, { \displaystyle -1 < x < 1 } , prove that { \displaystyle \frac{dy}{dx} = - \frac{1}{(1 + x)^2} }

Question 15. If { \displaystyle (x - a)^2 + (y - b)^2 = c^2 } , for some { \displaystyle c > 0 } , prove that { \displaystyle \frac{\left [ 1 + \left ( \frac{dy}{dx} \right )^2 \right ]^{\frac{3}{2}} }{\frac{d^2y}{dx^2}} } is a constant independent of { \displaystyle a } and { \displaystyle b } .

Question 16. If { \displaystyle \cos y = x \cos (a + y) } , with { \displaystyle \cos a \ne \pm 1 } , prove that { \displaystyle \frac{dy}{dx} = \frac{\cos^2(a + y)}{\sin a}} .

Question 17. If { \displaystyle x = a(\cos t + t \sin t) } and { \displaystyle y = a (\sin t - t \cos t)} , find { \displaystyle \frac{d^2y}{dx^2} } .

Question 19. Using mathematical induction prove that { \displaystyle \frac{d}{dx}(x^n) = nx^{n-1} } for all positive integers { \displaystyle n } .

Question 20. Using the fact that { \displaystyle \sin (A + B) = \sin A \cos B } { \displaystyle + \cos A \sin B } and the differentiation, obtain the sum formula for cosines.

Question 22. If { \displaystyle y = \begin{vmatrix} f(x) & g(x) & h(x) \\ l & m & n \\ a & b & c \end{vmatrix} } , prove that { \displaystyle \frac{dy}{dx} = \begin{vmatrix} f'(x) & g'(x) & h'(x) \\ l & m & n \\ a & b & c \end{vmatrix} } .

Question 23. If { \displaystyle y = e^{a \cos^{-1} x} } , { \displaystyle -1 \le x \le 1 } , show that { \displaystyle (1 - x^2)\frac{d^2y}{dx^2} } { \displaystyle - x \frac{dy}{dx} - a^2y = 0 } .