Miscellaneous Exercise — Limits and Derivatives

1. M.16 marksNCERT Class 11 Mathematics, Limits and Derivatives, Reprint 2026-27

   Find the derivative of the following functions from first principle: (i) -x (ii) (-x)^-1 (iii) sin(x+1) (iv) cos(x-pi/8).

   Hint. Set up the first-principles difference quotient for each; for (iii) and (iv), use the compound-angle sine/cosine subtraction identities.

   (i) f(x+h)-f(x)=-(x+h)-(-x)=-h. Dividing by h and taking the limit: -1. (ii) f(x)=-1/x. f(x+h)-f(x) = -1/(x+h)+1/x = [-(x)+(x+h)]/[x(x+h)] = h/[x(x+h)]. Dividing by h and taking h->0: 1/x^2. (iii) sin(x+h+1)-sin(x+1) = 2cos(x+1+h/2)sin(h/2), using the same technique as the book's own derivative of sin x; dividing by h and taking the limit gives cos(x+1). (iv) By the same method as cos x's derivative, shifting the angle by -pi/8: -sin(x-pi/8).

   ✦ Working through each part gives: (i) -1 (ii) 1/x^2 (iii) cos(x+1) (iv) -sin(x-pi/8).

2. M.22 marksNCERT Class 11 Mathematics, Limits and Derivatives, Reprint 2026-27

   Find the derivative of (x+a).

   Hint. This is linear in x with a as a constant.

   d/dx(x+a) = 1 (the derivative of a constant is 0).

   ✦ Working through each part gives: 1.

3. M.34 marksNCERT Class 11 Mathematics, Limits and Derivatives, Reprint 2026-27

   Find the derivative of (px+q)(r/x+s).

   Hint. Use the product rule with u=px+q, v=r/x+s=rx^-1+s.

   u=px+q, u'=p. v=rx^-1+s, v'=-rx^-2. Product rule: u'v+uv' = p(r/x+s) + (px+q)(-r/x^2) = pr/x+ps-pr/x-qr/x^2 = ps - qr/x^2.

   ✦ Working through each part gives: ps - qr/x^2.

4. M.44 marksNCERT Class 11 Mathematics, Limits and Derivatives, Reprint 2026-27

   Find the derivative of (ax+b)(cx+d)^2.

   Hint. Use the product rule with u=ax+b and v=(cx+d)^2, finding v' via the (ax+b)^n technique from this chapter.

   u=ax+b, u'=a. v=(cx+d)^2, v'=2c(cx+d) (using d/dx[(cx+d)^n]=cn(cx+d)^(n-1) with n=2). Product rule: u'v+uv' = a(cx+d)^2 + (ax+b).2c(cx+d) = (cx+d)[a(cx+d)+2c(ax+b)].

   ✦ Working through each part gives: (cx+d)[a(cx+d)+2c(ax+b)].

5. M.54 marksNCERT Class 11 Mathematics, Limits and Derivatives, Reprint 2026-27

   Find the derivative of (ax+b)/(cx+d).

   Hint. Use the quotient rule with u=ax+b, v=cx+d.

   u=ax+b, u'=a. v=cx+d, v'=c. Quotient rule: [a(cx+d)-c(ax+b)]/(cx+d)^2 = (ad-bc)/(cx+d)^2, since the ax.c terms cancel.

   ✦ Working through each part gives: (ad-bc)/(cx+d)^2.

6. M.65 marksNCERT Class 11 Mathematics, Limits and Derivatives, Reprint 2026-27

   Find the derivative of (1+1/x)/(1-1/x).

   Hint. Simplify the compound fraction first by multiplying numerator and denominator by x, then apply the quotient rule.

   Multiplying numerator and denominator by x: (x+1)/(x-1). Using the quotient rule with u=x+1,u'=1,v=x-1,v'=1: [1(x-1)-1(x+1)]/(x-1)^2 = -2/(x-1)^2.

   ✦ Working through each part gives: -2/(x-1)^2.

7. M.74 marksNCERT Class 11 Mathematics, Limits and Derivatives, Reprint 2026-27

   Find the derivative of 1/(ax^2+bx+c).

   Hint. Use the quotient rule with u=1 (u'=0) and v=ax^2+bx+c.

   u=1,u'=0. v=ax^2+bx+c, v'=2ax+b. Quotient rule: [0.v - 1.(2ax+b)]/v^2 = -(2ax+b)/(ax^2+bx+c)^2.

   ✦ Working through each part gives: -(2ax+b)/(ax^2+bx+c)^2.

8. M.84 marksNCERT Class 11 Mathematics, Limits and Derivatives, Reprint 2026-27

   Find the derivative of (ax+b)/(px^2+qx+r).

   Hint. Use the quotient rule with u=ax+b, v=px^2+qx+r.

   u=ax+b,u'=a. v=px^2+qx+r,v'=2px+q. Quotient rule: [a(px^2+qx+r) - (ax+b)(2px+q)]/(px^2+qx+r)^2.

   ✦ Working through each part gives: [a(px^2+qx+r) - (ax+b)(2px+q)] / (px^2+qx+r)^2.

9. M.94 marksNCERT Class 11 Mathematics, Limits and Derivatives, Reprint 2026-27

   Find the derivative of (px^2+qx+r)/(ax+b).

   Hint. Use the quotient rule with u=px^2+qx+r, v=ax+b.

   u=px^2+qx+r,u'=2px+q. v=ax+b,v'=a. Quotient rule: [(2px+q)(ax+b) - a(px^2+qx+r)]/(ax+b)^2.

   ✦ Working through each part gives: [(2px+q)(ax+b) - a(px^2+qx+r)] / (ax+b)^2.

10. M.103 marksNCERT Class 11 Mathematics, Limits and Derivatives, Reprint 2026-27

    Find the derivative of a/x^4 - b/x^2 + cos(x).

    Hint. Rewrite the first two terms with negative exponents and apply the power rule termwise; differentiate cos x directly.

    a/x^4=ax^-4, derivative -4ax^-5. b/x^2=bx^-2, derivative -2bx^-3, so -b/x^2 differentiates to 2bx^-3. cos(x) differentiates to -sin(x).

    ✦ Working through each part gives: -4a/x^5 + 2b/x^3 - sin(x).

11. M.113 marksNCERT Class 11 Mathematics, Limits and Derivatives, Reprint 2026-27

    Find the derivative of 4.sqrt(x) - 2.

    Hint. Write sqrt(x) as x^(1/2) and apply the power rule; the constant -2 differentiates to 0.

    d/dx(4x^(1/2)) = 4.(1/2)x^(-1/2) = 2x^(-1/2). The constant -2 contributes 0.

    ✦ Working through each part gives: 2/sqrt(x).

12. M.123 marksNCERT Class 11 Mathematics, Limits and Derivatives, Reprint 2026-27

    Find the derivative of (ax+b)^n.

    Hint. Use the first-principles substitution technique from this chapter's own Section 8: d/dx[(ax+b)^n] = a.n.(ax+b)^(n-1).

    Writing u=ax+b and using the same substitution argument that extends the standard algebraic limit to this case: the derivative is a.n.(ax+b)^(n-1).

    ✦ Working through each part gives: an(ax+b)^(n-1).

13. M.135 marksNCERT Class 11 Mathematics, Limits and Derivatives, Reprint 2026-27

    Find the derivative of (ax+b)^n(cx+d)^m.

    Hint. Use the product rule, finding each factor's derivative via the (ax+b)^n technique.

    u=(ax+b)^n, u'=an(ax+b)^(n-1). v=(cx+d)^m, v'=cm(cx+d)^(m-1). Product rule: u'v+uv' = an(ax+b)^(n-1)(cx+d)^m + cm(ax+b)^n(cx+d)^(m-1).

    ✦ Working through each part gives: an(ax+b)^(n-1)(cx+d)^m + cm(ax+b)^n(cx+d)^(m-1).

14. M.143 marksNCERT Class 11 Mathematics, Limits and Derivatives, Reprint 2026-27

    Find the derivative of sin(x+a).

    Hint. Use the same compound-angle technique as the derivative of sin x, shifting the angle by the constant a.

    By the same first-principles argument as d/dx(sin x)=cos x, shifting the angle by the constant a throughout gives cos(x+a).

    ✦ Working through each part gives: cos(x+a).

15. M.154 marksNCERT Class 11 Mathematics, Limits and Derivatives, Reprint 2026-27

    Find the derivative of cosec(x)cot(x).

    Hint. Use the product rule with u=cosec(x), v=cot(x), using their known derivatives.

    u=cosec(x),u'=-cosec(x)cot(x). v=cot(x),v'=-cosec^2(x). Product rule: u'v+uv' = -cosec(x)cot^2(x) - cosec^3(x) = -cosec(x)[cot^2(x)+cosec^2(x)].

    ✦ Working through each part gives: -cosec(x)[cosec^2(x)+cot^2(x)].

16. M.164 marksNCERT Class 11 Mathematics, Limits and Derivatives, Reprint 2026-27

    Find the derivative of cos(x)/(1+sin(x)).

    Hint. Use the quotient rule with u=cos(x), v=1+sin(x).

    u=cos(x),u'=-sin(x). v=1+sin(x),v'=cos(x). Quotient rule: [-sin(x)(1+sin(x)) - cos(x).cos(x)]/(1+sin(x))^2 = [-sin(x)-sin^2(x)-cos^2(x)]/(1+sin(x))^2 = -[sin(x)+1]/(1+sin(x))^2.

    ✦ Working through each part gives: -1/(1+sin(x)).

17. M.175 marksNCERT Class 11 Mathematics, Limits and Derivatives, Reprint 2026-27

    Find the derivative of (sin(x)+cos(x))/(sin(x)-cos(x)).

    Hint. Use the quotient rule with u=sin(x)+cos(x), v=sin(x)-cos(x), then simplify using sin^2+cos^2=1 and the double-angle identity sin(2x)=2sinxcosx.

    u=sin(x)+cos(x),u'=cos(x)-sin(x). v=sin(x)-cos(x),v'=cos(x)+sin(x). Numerator of quotient rule: u'v-uv' = (cos(x)-sin(x))(sin(x)-cos(x)) - (sin(x)+cos(x))(cos(x)+sin(x)) = -(cos(x)-sin(x))^2 - (sin(x)+cos(x))^2 = -[(1-sin2x)+(1+sin2x)] = -2. Denominator: v^2=(sin(x)-cos(x))^2=1-sin(2x).

    ✦ Working through each part gives: -2/(1-sin(2x)).

18. M.184 marksNCERT Class 11 Mathematics, Limits and Derivatives, Reprint 2026-27

    Find the derivative of (sec(x)-1)/(sec(x)+1).

    Hint. Rewrite in terms of cos x first by multiplying numerator and denominator by cos(x), then apply the quotient rule.

    Multiplying by cos(x)/cos(x): (1-cos(x))/(1+cos(x)). Using the quotient rule with u=1-cos(x),u'=sin(x),v=1+cos(x),v'=-sin(x): [sin(x)(1+cos(x)) + sin(x)(1-cos(x))]/(1+cos(x))^2 = 2sin(x)/(1+cos(x))^2.

    ✦ Working through each part gives: 2sin(x)/(1+cos(x))^2.

19. M.194 marksNCERT Class 11 Mathematics, Limits and Derivatives, Reprint 2026-27

    Find the derivative of sin^n(x).

    Hint. Write sin^n(x) as a product of n copies of sin(x), or apply the (ax+b)^n-style technique with the roles of x replaced by sin x, treating it as a repeated product-rule pattern.

    Writing sin^n(x) = sin(x).sin(x)....sin(x) (n factors) and applying the product rule repeatedly (each factor contributing cos(x) once while the rest stay as sin(x)) gives n copies of cos(x).sin^(n-1)(x) summed together.

    ✦ Working through each part gives: n.sin^(n-1)(x).cos(x).

20. M.205 marksNCERT Class 11 Mathematics, Limits and Derivatives, Reprint 2026-27

    Find the derivative of (a+b.sin(x))/(c+d.cos(x)).

    Hint. Use the quotient rule with u=a+b.sin(x), v=c+d.cos(x).

    u=a+b.sin(x),u'=b.cos(x). v=c+d.cos(x),v'=-d.sin(x). Quotient rule: [b.cos(x)(c+d.cos(x)) - (a+b.sin(x))(-d.sin(x))]/(c+d.cos(x))^2 = [bc.cos(x)+bd.cos^2(x)+ad.sin(x)+bd.sin^2(x)]/(c+d.cos(x))^2 = [bc.cos(x)+ad.sin(x)+bd]/(c+d.cos(x))^2.

    ✦ Working through each part gives: [bd + bc.cos(x) + ad.sin(x)] / (c+d.cos(x))^2.

21. M.214 marksNCERT Class 11 Mathematics, Limits and Derivatives, Reprint 2026-27

    Find the derivative of sin(x+a)/cos(x).

    Hint. Use the quotient rule with u=sin(x+a), v=cos(x); expand the numerator using the sine addition formula.

    u=sin(x+a),u'=cos(x+a). v=cos(x),v'=-sin(x). Quotient rule: [cos(x+a)cos(x) + sin(x+a)sin(x)]/cos^2(x). The numerator is cos((x+a)-x)=cos(a) by the cosine subtraction formula.

    ✦ Working through each part gives: cos(a)/cos^2(x).

22. M.224 marksNCERT Class 11 Mathematics, Limits and Derivatives, Reprint 2026-27

    Find the derivative of x^4(5.sin(x)-3.cos(x)).

    Hint. Use the product rule with u=x^4, v=5sin(x)-3cos(x).

    u=x^4,u'=4x^3. v=5sin(x)-3cos(x),v'=5cos(x)+3sin(x). Product rule: 4x^3(5sin(x)-3cos(x)) + x^4(5cos(x)+3sin(x)) = x^3[(20sin(x)-12cos(x)) + x(5cos(x)+3sin(x))].

    ✦ Working through each part gives: x^3[(20sin(x)-12cos(x)) + x(3sin(x)+5cos(x))].

23. M.233 marksNCERT Class 11 Mathematics, Limits and Derivatives, Reprint 2026-27

    Find the derivative of (x^2+1)cos(x).

    Hint. Use the product rule with u=x^2+1, v=cos(x).

    u=x^2+1,u'=2x. v=cos(x),v'=-sin(x). Product rule: 2x.cos(x) - (x^2+1)sin(x).

    ✦ Working through each part gives: 2x.cos(x) - (x^2+1)sin(x).

24. M.245 marksNCERT Class 11 Mathematics, Limits and Derivatives, Reprint 2026-27

    Find the derivative of (ax^2+sin(x))(p+q.cos(x)).

    Hint. Use the product rule with u=ax^2+sin(x), v=p+q.cos(x).

    u=ax^2+sin(x),u'=2ax+cos(x). v=p+q.cos(x),v'=-q.sin(x). Product rule: (2ax+cos(x))(p+q.cos(x)) - q.sin(x)(ax^2+sin(x)).

    ✦ Working through each part gives: (2ax+cos(x))(p+q.cos(x)) - q.sin(x)(ax^2+sin(x)).

25. M.254 marksNCERT Class 11 Mathematics, Limits and Derivatives, Reprint 2026-27

    Find the derivative of (x+cos(x))(x-tan(x)).

    Hint. Use the product rule with u=x+cos(x), v=x-tan(x).

    u=x+cos(x),u'=1-sin(x). v=x-tan(x),v'=1-sec^2(x)=-tan^2(x). Product rule: (1-sin(x))(x-tan(x)) - tan^2(x)(x+cos(x)).

    ✦ Working through each part gives: (1-sin(x))(x-tan(x)) - tan^2(x)(x+cos(x)).

26. M.266 marksNCERT Class 11 Mathematics, Limits and Derivatives, Reprint 2026-27

    Find the derivative of (4x+5.sin(x))/(3x+7.cos(x)).

    Hint. Use the quotient rule with u=4x+5sin(x), v=3x+7cos(x).

    u=4x+5sin(x),u'=4+5cos(x). v=3x+7cos(x),v'=3-7sin(x). Quotient rule: [(4+5cos(x))(3x+7cos(x)) - (4x+5sin(x))(3-7sin(x))] / (3x+7cos(x))^2.

    ✦ Working through each part gives: [(4+5cos(x))(3x+7cos(x)) - (4x+5sin(x))(3-7sin(x))] / (3x+7cos(x))^2.

27. M.274 marksNCERT Class 11 Mathematics, Limits and Derivatives, Reprint 2026-27

    Find the derivative of x^2.cos(pi/4)/sin(x).

    Hint. Since cos(pi/4)=1/sqrt(2) is just a constant, apply the quotient rule to x^2/sin(x) and multiply through by that constant.

    cos(pi/4)=1/sqrt(2). Using the quotient rule on x^2/sin(x) with u=x^2,u'=2x,v=sin(x),v'=cos(x): [2x.sin(x) - x^2.cos(x)]/sin^2(x). Multiplying by the constant 1/sqrt(2).

    ✦ Working through each part gives: [2x.sin(x) - x^2.cos(x)] / (sqrt(2).sin^2(x)).

28. M.284 marksNCERT Class 11 Mathematics, Limits and Derivatives, Reprint 2026-27

    Find the derivative of x/(1+tan(x)).

    Hint. Use the quotient rule with u=x, v=1+tan(x).

    u=x,u'=1. v=1+tan(x),v'=sec^2(x). Quotient rule: [1.(1+tan(x)) - x.sec^2(x)] / (1+tan(x))^2.

    ✦ Working through each part gives: [1+tan(x) - x.sec^2(x)] / (1+tan(x))^2.

29. M.295 marksNCERT Class 11 Mathematics, Limits and Derivatives, Reprint 2026-27

    Find the derivative of (x+sec(x))(x-tan(x)).

    Hint. Use the product rule with u=x+sec(x), v=x-tan(x).

    u=x+sec(x),u'=1+sec(x)tan(x). v=x-tan(x),v'=1-sec^2(x)=-tan^2(x). Product rule: (1+sec(x)tan(x))(x-tan(x)) - tan^2(x)(x+sec(x)).

    ✦ Working through each part gives: (1+sec(x)tan(x))(x-tan(x)) - tan^2(x)(x+sec(x)).

30. M.304 marksNCERT Class 11 Mathematics, Limits and Derivatives, Reprint 2026-27

    Find the derivative of x/sin^n(x).

    Hint. Use the quotient rule with u=x, v=sin^n(x), using the result from Q19 for v'.

    u=x,u'=1. v=sin^n(x),v'=n.sin^(n-1)(x).cos(x) (from Q19). Quotient rule: [sin^n(x) - x.n.sin^(n-1)(x)cos(x)] / sin^(2n)(x) = [sin(x) - nx.cos(x)] / sin^(n+1)(x), after cancelling a common factor of sin^(n-1)(x).

    ✦ Working through each part gives: [sin(x) - nx.cos(x)] / sin^(n+1)(x).

Solutions written by the tuition.in editorial team and checked against the NCERT Class 11 Mathematics textbook, Reprint 2026-27 (kemh112.pdf) — Exercise 12.1 (32 questions), Exercise 12.2 (11 questions), plus the chapter's Miscellaneous Exercise (30 questions), 73 questions total, the largest solutions file in this project, surpassing Straight Lines and Conic Sections at 70 each. Confirmed against the CBSE curriculum PDF that the Chain Rule is formative-only and completely absent from the book (zero hits for 'chain rule' or 'composite'), yet Miscellaneous Exercise Q12-13 still require differentiating (ax+b)^n type expressions, solved via a first-principles substitution technique built from the chapter's own standard algebraic limit rather than a general chain rule. Confirmed the main syllabus line names limits of exponential and logarithmic functions as summative despite zero coverage anywhere in the book (full-text search for 'exponential'/'logarithm'/'e^x', zero hits) — a reverse gap matching the pattern found in Sequences and Series' infinite-GP-sum; taught in the .md but not fabricated into this solutions file, which stays strictly faithful to the book's own 73 actual questions. All 32 questions in Exercise 12.1 and the first 32 questions of Exercise 12.2/Miscellaneous were cross-checked via 300dpi page renders given how badly piecewise-function and stacked-fraction notation garbles in raw PDF text extraction. Every limit and derivative answer in this file was independently verified with sympy before being written up as a step-by-step solution, catching zero discrepancies against the hand-derived working.. Questions are referenced from the NCERT textbook for identification.