NCERT Solutions for Class 11 Maths – Chapter 12 Limits and Derivatives Miscellaneous Exercise

NCERT Solutions for Class 11 Maths Chapter 12 Miscellaneous Exercise – Limits and Derivatives covers advanced problems on finding derivatives using the first principle and standard rules. This exercise includes derivatives of algebraic, trigonometric, and composite functions, testing students' complete understanding of the chapter.

Prepared according to the latest CBSE Class 11 Maths syllabus 2025-26, this Miscellaneous Exercise is important for board exam preparation as it combines multiple concepts — first principle, product rule, quotient rule, and standard trig derivatives — in a single set of problems.

NCERT Solutions for Class 11 Maths – Chapter 12 Limits and Derivatives Miscellaneous Exercise

Limits and Derivatives

Medium

Q.

Find the derivative of the following functions from first principle.

$\begin{array}{l}\mathbf{\left(}\mathbf{i}\mathbf{\right)}\mathbf{ }{\mathbf{x}}^{\mathbf{3}}\mathbf{-}\mathbf{27}\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{\left(}\mathbf{ii}\mathbf{\right)}\mathbf{ }\mathbf{\left(}\mathbf{x}\mathbf{-}\mathbf{1}\mathbf{\right)}\mathbf{\left(}\mathbf{x}\mathbf{-}\mathbf{2}\mathbf{\right)}\\ \mathbf{\left(}\mathbf{iii}\mathbf{\right)}\mathbf{ }\frac{\mathbf{1}}{{\mathbf{x}}^{\mathbf{2}}}\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{\left(}\mathbf{iv}\mathbf{\right)}\mathbf{ }\frac{\mathbf{x}\mathbf{+}\mathbf{1}}{\mathbf{x}\mathbf{-}\mathbf{1}}\end{array}$

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Limits and Derivatives

Difficult

Q.

$\begin{array}{l}\mathbf{Find}\mathbf{ }\mathbf{the}\mathbf{ }\mathbf{derivative}\mathbf{ }\mathbf{of}\\ \mathbf{\left(}\mathbf{i}\mathbf{\right)}\mathbf{ }\mathbf{2}\mathbf{x}\mathbf{-}\frac{\mathbf{3}}{\mathbf{4}}\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{\left(}\mathbf{ii}\mathbf{\right)}\mathbf{ }\mathbf{\left(}\mathbf{5}{\mathbf{x}}^{\mathbf{3}}\mathbf{+}\mathbf{3}\mathbf{x}\mathbf{-}\mathbf{1}\mathbf{\right)}\mathbf{\left(}\mathbf{x}\mathbf{-}\mathbf{1}\mathbf{\right)}\\ \mathbf{\left(}\mathbf{iii}\mathbf{\right)}\mathbf{ }{\mathbf{x}}^{\mathbf{-}\mathbf{3}}\mathbf{\left(}\mathbf{5}\mathbf{+}\mathbf{3}\mathbf{x}\mathbf{\right)}\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{\left(}\mathbf{iv}\mathbf{\right)}\mathbf{ }{\mathbf{x}}^{\mathbf{5}}\mathbf{\left(}\mathbf{3}\mathbf{-}\mathbf{6}{\mathbf{x}}^{\mathbf{-}\mathbf{9}}\mathbf{\right)}\\ \mathbf{\left(}\mathbf{v}\mathbf{\right)}\mathbf{ }{\mathbf{x}}^{\mathbf{-}\mathbf{4}}\mathbf{\left(}\mathbf{3}\mathbf{-}\mathbf{4}{\mathbf{x}}^{\mathbf{-}\mathbf{5}}\mathbf{\right)}\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{\left(}\mathbf{vi}\mathbf{\right)}\mathbf{ }\frac{\mathbf{2}}{\mathbf{x}\mathbf{+}\mathbf{1}}\mathbf{-}\frac{{\mathbf{x}}^{\mathbf{2}}}{\mathbf{3}\mathbf{x}\mathbf{-}\mathbf{1}}\end{array}$

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Limits and Derivatives

Medium

Q.

Find the derivative of the following functions:

(i) sin x cos x
(ii) sec x
(iii) 5sec x + 4cos x
(iv) cosec x
(v) 3cot x + 5cosec x
(vi) 5sin x − 6cos x + 7
(vii) 2 tan x − 7sec x

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Limits and Derivatives

Easy

Q.

$\underset{\mathbf{x}\mathbf{\to }\mathbf{0}}{\mathbf{lim}}\frac{{\mathbf{e}}^{\mathbf{4}\mathbf{x}}\mathbf{-}\mathbf{1}}{\mathbf{x}}$

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Limits and Derivatives

Difficult

Q.

Find the derivative of the following functions from first principle:
$\mathbf{\left(}\mathbf{i}\mathbf{\right)}\mathbf{ }\mathbf{-}\mathbf{x}\mathbf{ }\mathbf{\left(}\mathbf{ii}\mathbf{\right)}\mathbf{ }{\mathbf{(}\mathbf{-}\mathbf{x}\mathbf{)}}^{\mathbf{-}\mathbf{1}}\mathbf{ }\mathbf{\left(}\mathbf{iii}\mathbf{\right)}\mathbf{ }\mathbf{sin}\mathbf{(}\mathbf{x}\mathbf{+}\mathbf{1}\mathbf{)}\mathbf{ }\mathbf{\left(}\mathbf{iv}\mathbf{\right)}\mathbf{ }\mathbf{cos}\mathbf{(}\mathbf{x}\mathbf{-}\frac{\mathbf{\pi }}{\mathbf{8}}\mathbf{)}$

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Limits and Derivatives

Easy

Q.

Find the derivative of the function (x + a).

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Limits and Derivatives

Easy

Q.

$\mathbf{\text{Find the derivative of the function }}\frac{\mathbf{\text{ax+b}}}{\mathbf{\text{cx+d}}}\mathbf{.}$

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Limits and Derivatives

Easy

Q.

$\mathbf{\text{Find the derivative of the function }}\frac{\mathbf{\text{1+}}\frac{\mathbf{\text{1}}}{\mathbf{\text{x}}}}{\mathbf{\text{1}}\mathbf{-}\frac{\mathbf{\text{1}}}{\mathbf{\text{x}}}}\mathbf{.}$

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Limits and Derivatives

Easy

Q.

$\mathbf{\text{Find the derivative of the function }}\frac{{\mathbf{\text{px}}}^{\mathbf{\text{2}}}\mathbf{\text{+ qx + r}}}{\mathbf{\text{ax + b}}}\mathbf{.}$

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Limits and Derivatives

Easy

Q.

$\mathbf{\text{Find the derivative of the function }}\frac{\mathbf{\text{a}}}{{\mathbf{\text{x}}}^{\mathbf{\text{4}}}}\mathbf{-}\frac{\mathbf{\text{b}}}{{\mathbf{\text{x}}}^{\mathbf{\text{2}}}}\mathbf{\text{+cos x}}\mathbf{.}$

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Limits and Derivatives

Easy

Q.

$\mathbf{\text{Find the derivative of the function 4}}\sqrt{\mathbf{\text{x}}}\mathbf{-}\mathbf{\text{2.}}$

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Limits and Derivatives

Medium

Q.

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

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Limits and Derivatives

Difficult

Q.

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

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The solutions are explained in a clear, step-by-step format to help students revise and strengthen their understanding of limits and derivatives.

Q1. Find the derivative of the following functions from first principle:

(i) −x

Ans: Let f(x) = −x, so f(x+h) = −(x+h)

f'(x) = lim(h→0) [f(x+h) − f(x)] / h = lim(h→0) [−(x+h) − (−x)] / h

= lim(h→0) [−x − h + x] / h = lim(h→0) (−h/h) = −1

(ii) (−x)⁻¹

Ans: Let f(x) = (−x)⁻¹ = −1/x, so f(x+h) = −1/(x+h)

f'(x) = lim(h→0) [−1/(x+h) − (−1/x)] / h

= lim(h→0) [−x + (x+h)] / [h·x(x+h)]

= lim(h→0) h / [h·x(x+h)]

= lim(h→0) 1/[x(x+h)] = 1/x²

(iii) sin(x+1)

Ans: Let f(x) = sin(x+1), so f(x+h) = sin(x+h+1)

f'(x) = lim(h→0) [sin(x+h+1) − sin(x+1)] / h

Using sin A − sin B = 2 cos((A+B)/2) sin((A−B)/2):

= lim(h→0) [2 cos(x+1+h/2) · sin(h/2)] / h

= lim(h→0) cos(x+1+h/2) · [sin(h/2)/(h/2)]

= cos(x+1) × 1 = cos(x+1)

(iv) cos(π/8 − x)

Ans: Let f(x) = cos(π/8 − x), so f(x+h) = cos(π/8 − x − h)

f'(x) = lim(h→0) [cos(π/8−x−h) − cos(π/8−x)] / h

Using cos A − cos B = −2 sin((A+B)/2) sin((A−B)/2):

= lim(h→0) [−2 sin(π/8−x−h/2) · sin(−h/2)] / h

= lim(h→0) sin(π/8−x−h/2) · [sin(h/2)/(h/2)]

= sin(π/8−x) × 1 = sin(π/8 − x)

Q2. Find the derivative of the following functions:

(i) sin x + cos x

Ans: d/dx(sin x) + d/dx(cos x) = cos x + (−sin x) = cos x − sin x

(ii) asin x + bcos x

Ans: f'(x) = a cos x − b sin x

(iii) x/(sin x + cos x)

Ans: By quotient rule: f'(x) = [(sin x + cos x)·1 − x·(cos x − sin x)] / (sin x + cos x)²

= [sin x + cos x − x cos x + x sin x] / (sin x + cos x)²

f'(x) = [sin x + cos x + x(sin x − cos x)] / (sin x + cos x)²

(iv) (a + b sin x)/(c + d cos x)

Ans: By quotient rule:

f'(x) = [(c + d cos x)(b cos x) − (a + b sin x)(−d sin x)] / (c + d cos x)²

= [bc cos x + bd cos²x + ad sin x + bd sin²x] / (c + d cos x)²

= [bc cos x + ad sin x + bd(sin²x + cos²x)] / (c + d cos x)²

f'(x) = (bc cos x + ad sin x + bd) / (c + d cos x)²

(v) (sin x − cos x)/(sin x + cos x)

Ans: By quotient rule:

f'(x) = [(sin x + cos x)(cos x + sin x) − (sin x − cos x)(cos x − sin x)] / (sin x + cos x)²

Numerator = (sin x + cos x)² − (sin x − cos x)(cos x − sin x)

= (sin x + cos x)² + (sin x − cos x)² = 1 + 1 = 2

f'(x) = 2/(sin x + cos x)²

(vi) (a + bsin x)²

Ans: f(x) = a² + 2ab sin x + b² sin²x

f'(x) = 2ab cos x + b²·2 sin x·cos x = 2ab cos x + b² sin 2x

f'(x) = 2b(a cos x + b sin x cos x) = 2b cos x(a + b sin x)

(vii) sin²x

Ans: f(x) = sin²x

f'(x) = 2 sin x · cos x = sin 2x

(viii) (2x − 7)²(3x − 5)

Ans: By Leibnitz product rule:

Let u = (2x−7)², v = (3x−5)

u' = 2(2x−7)·2 = 4(2x−7), v' = 3

f'(x) = 4(2x−7)(3x−5) + (2x−7)²·3

= (2x−7)[4(3x−5) + 3(2x−7)]

= (2x−7)[12x − 20 + 6x − 21]

= (2x−7)(18x − 41)

f'(x) = (2x−7)(18x−41)

(ix) (5x + 3)/(x² + 4)

Ans: By quotient rule:

f'(x) = [(x²+4)(5) − (5x+3)(2x)] / (x²+4)²

= [5x² + 20 − 10x² − 6x] / (x²+4)²

f'(x) = (−5x² − 6x + 20) / (x²+4)²

(x) (x² + 1) cos x

Ans: By Leibnitz product rule:

f'(x) = (x²+1)(−sin x) + cos x·(2x)

f'(x) = 2x cos x − (x²+1) sin x

(xi) (ax² + sin x)(p + q cos x)

Ans: By Leibnitz product rule:

f'(x) = (ax²+sin x)(−q sin x) + (p+q cos x)(2ax+cos x)

= −aq x² sin x − q sin²x + 2apx + p cos x + 2aqx cos x + q cos²x

f'(x) = (p+q cos x)(2ax+cos x) − q sin x(ax²+sin x)

(xii) (x + cos x)(x − tan x)

Ans: By Leibnitz product rule:

f'(x) = (x+cos x)(1−sec²x) + (x−tan x)(1−sin x)

f'(x) = (x+cos x)(1−sec²x) + (x−tan x)(1−sin x)

(xiii) 4√x − 2

Ans: f(x) = 4x^(1/2) − 2

f'(x) = 4·(1/2)x^(−1/2) = 2/√x

f'(x) = 2/√x

(xiv) (ax + b)ⁿ

Ans: Using chain rule: f'(x) = n(ax+b)^(n−1)·a

f'(x) = an(ax+b)^(n−1)

(xv) (ax + b)ⁿ(cx + d)ᵐ

Ans: By Leibnitz product rule:

f'(x) = (ax+b)ⁿ·m(cx+d)^(m−1)·c + (cx+d)ᵐ·n(ax+b)^(n−1)·a

= (ax+b)^(n−1)(cx+d)^(m−1)[mc(ax+b) + na(cx+d)]

f'(x) = (ax+b)^(n−1)(cx+d)^(m−1)[mc(ax+b) + na(cx+d)]

(xvi) sin(x+a)/cos x

Ans: f(x) = sin(x+a)/cos x = [sin x·cos a + cos x·sin a]/cos x = cos a·tan x + sin a

f'(x) = cos a·sec²x

f'(x) = cos a · sec²x

(xvii) cosec x·cot x

Ans: By Leibnitz product rule:

f'(x) = cosec x·(−cosec²x) + cot x·(−cosec x·cot x)

= −cosec³x − cosec x·cot²x

= −cosec x(cosec²x + cot²x)

f'(x) = −cosec x(cosec²x + cot²x)

(xviii) (tan x − 1)/sec x

Ans: f(x) = (tan x − 1)/sec x = sin x − cos x

f'(x) = cos x + sin x

f'(x) = cos x + sin x

(xix) (x² + 1)(x + 5) + (x³ + 2)

Ans: f(x) = x³ + 5x² + x + 5 + x³ + 2 = 2x³ + 5x² + x + 7

f'(x) = 6x² + 10x + 1

(xx) (3x + 1)/x²

Ans: f(x) = 3/x + 1/x² = 3x⁻¹ + x⁻²

f'(x) = −3x⁻² − 2x⁻³

f'(x) = −3/x² − 2/x³

FAQs – Class 11 Maths Chapter 12 Miscellaneous Exercise Limits and Derivatives

Q1. What is covered in the Miscellaneous Exercise of Chapter 12? The Miscellaneous Exercise covers derivatives of algebraic and trigonometric functions using first principles, product rule, quotient rule, and standard formulas. It is a comprehensive revision of the entire chapter.

Q2. Which functions are differentiated using the first principle in this exercise? In Q1, the functions −x, (−x)⁻¹, sin(x+1), and cos(π/8−x) are differentiated using the first principle (limit definition of derivative).

Q3. What is the Leibnitz product rule and when is it used? The product rule states: d/dx[u·v] = u'v + uv'. It is used whenever a function is expressed as a product of two differentiable functions.

Q4. What is the quotient rule? The quotient rule states: d/dx[u/v] = (u'v − uv')/v². It is used to differentiate functions expressed as a ratio of two functions.

Q5. Why is the Miscellaneous Exercise important for exams? This exercise combines all concepts of the chapter and contains question types that are frequently asked in CBSE board exams. It tests the ability to apply multiple differentiation techniques in a single problem.

Q6. How can students prepare effectively for the Miscellaneous Exercise? Students should revise all standard derivative formulas, practice first principle questions, and master the product and quotient rules. Solving all 20 sub-parts of Q2 is especially important for thorough preparation.