Rd Sharma Xi 2019 Solutions for Class 12 Commerce Maths Chapter 30 Derivatives are provided here with simple step-by-step explanations. These solutions for Derivatives are extremely popular among Class 12 Commerce students for Maths Derivatives Solutions come handy for quickly completing your homework and preparing for exams. All questions and answers from the Rd Sharma Xi 2019 Book of Class 12 Commerce Maths Chapter 30 are provided here for you for free. You will also love the ad-free experience on Meritnation’s Rd Sharma Xi 2019 Solutions. All Rd Sharma Xi 2019 Solutions for class Class 12 Commerce Maths are prepared by experts and are 100% accurate.

#### Question 1:

(i) $\frac{2}{x}$

(ii) $\frac{1}{\sqrt{x}}$

(iii) $\frac{1}{{x}^{3}}$

(iv) $\frac{{x}^{2}+1}{x}$

(v) $\frac{{x}^{2}-1}{x}$

(vi) $\frac{x+1}{x+2}$

(vii) $\frac{x+2}{3x+5}$

(viii) k xn

(ix) $\frac{1}{\sqrt{3-x}}$

(x) x2 + x + 3

(xi) (x + 2)3

(xii) x3 + 4x2 + 3x + 2

(xiii) (x2 + 1) (x − 5)

(xiv) $\sqrt{2{x}^{2}+1}$

(xv) $\frac{2x+3}{x-2}$

#### Question 2:

Differentiate each of the following from first principles:

(i) ex

(ii) e3x

(iii) eax + b

(iv) x ex

(v) − x

(vi) (−x)−1

(vii) sin (x + 1)

(viii) $\mathrm{cos}\left(x-\frac{\mathrm{\pi }}{8}\right)$

(ix) x sin x

(x) x cos x

(xi) sin (2x − 3)

#### Question 3:

Differentiate each of the following from first principles:

(i)

(ii)

(iii)

(iv) x2 sin x

(v)

(vi) sin x + cos x

(vii) x2 ex

(viii) ${e}^{{x}^{2}+1}$

(ix) ${e}^{\sqrt{2x}}$

(x) ${e}^{\sqrt{ax+b}}$

(xi) ${a}^{\sqrt{x}}$

(x) ${3}^{{x}^{2}}$

#### Question 4:

(i) tan2 x

(ii) tan (2x + 1)

(iii) tan 2x

(iv)

(i)
(ii)
(iii)
(iv) tan x2

#### Question 1:

Find the derivative of f (x) = 3x at x = 2

We have:

#### Question 2:

Find the derivative of f (x) = x2 − 2 at x = 10

We have:

#### Question 3:

Find the derivative of f (x) = 99x at x = 100

We have:

#### Question 4:

Find the derivative of f (x) x at x = 1

We have:

#### Question 5:

Find the derivative of f (x) = cos x at x = 0

We have:

#### Question 6:

Find the derivative of f (x) = tan x at x = 0

We have:

#### Question 7:

Find the derivatives of the following functions at the indicated points:

(i) sin x at x = $\frac{\mathrm{\pi }}{2}$

(ii) x at x = 1

(iii) 2 cos x at x = $\frac{\mathrm{\pi }}{2}$

(iv) sin 2x at x = $\frac{\mathrm{\pi }}{2}$

#### Question 1:

x4 − 2 sin x + 3 cos x

3x + x3 + 33

#### Question 3:

$\frac{{x}^{3}}{3}-2\sqrt{x}+\frac{5}{{x}^{2}}$

$\frac{d}{dx}\left(\frac{{x}^{3}}{3}-2\sqrt{x}+\frac{5}{{x}^{2}}\right)\phantom{\rule{0ex}{0ex}}=\frac{1}{3}\frac{d}{dx}\left({x}^{3}\right)-2\frac{d}{dx}\left({x}^{\frac{1}{2}}\right)+5\frac{d}{dx}\left({x}^{-2}\right)\phantom{\rule{0ex}{0ex}}=\frac{1}{3}\left(3{x}^{2}\right)-2.\frac{1}{2}.{x}^{\frac{-1}{2}}+5\left(-2\right){x}^{-3}\phantom{\rule{0ex}{0ex}}={x}^{2}-{x}^{\frac{-1}{2}}-10{x}^{-3}$

#### Question 4:

ex log a + ea long x + ea log a

#### Question 5:

(2x2 + 1) (3x + 2)

$\frac{d}{dx}\left(\left(2{x}^{2}+1\right)\left(3x+2\right)\right)\phantom{\rule{0ex}{0ex}}=\frac{d}{dx}\left(6{x}^{3}+4{x}^{2}+3x+2\right)\phantom{\rule{0ex}{0ex}}=6\frac{d}{dx}\left({x}^{3}\right)+4\frac{d}{dx}\left({x}^{2}\right)+3\frac{d}{dx}\left(x\right)+\frac{d}{dx}\left(2\right)\phantom{\rule{0ex}{0ex}}=6\left(3{x}^{2}\right)+4\left(2x\right)+3\left(1\right)+0\phantom{\rule{0ex}{0ex}}=18{x}^{2}+8x+3$

#### Question 6:

log3 x + 3 loge x + 2 tan x

#### Question 7:

$\left(x+\frac{1}{x}\right)\left(\sqrt{x}+\frac{1}{\sqrt{x}}\right)$

$\frac{d}{dx}\left[\left(x+\frac{1}{x}\right)\left(\sqrt{x}+\frac{1}{\sqrt{x}}\right)\right]\phantom{\rule{0ex}{0ex}}=\frac{d}{dx}\left[\left(x+{x}^{-1}\right)\left({x}^{\frac{1}{2}}+{x}^{\frac{-1}{2}}\right)\right]\phantom{\rule{0ex}{0ex}}=\frac{d}{dx}\left({x}^{\frac{3}{2}}+{x}^{\frac{1}{2}}+{x}^{\frac{-1}{2}}+{x}^{\frac{-3}{2}}\right)\phantom{\rule{0ex}{0ex}}=\frac{d}{dx}\left({x}^{\frac{3}{2}}\right)+\frac{d}{dx}\left({x}^{\frac{1}{2}}\right)+\frac{d}{dx}\left({x}^{\frac{-1}{2}}\right)+\frac{d}{dx}\left({x}^{\frac{-3}{2}}\right)\phantom{\rule{0ex}{0ex}}=\frac{3}{2}{x}^{\frac{1}{2}}+\frac{1}{2}{x}^{\frac{-1}{2}}-\frac{1}{2}{x}^{\frac{-3}{2}}-\frac{3}{2}{x}^{\frac{-5}{2}}\phantom{\rule{0ex}{0ex}}$

#### Question 8:

${\left(\sqrt{x}+\frac{1}{\sqrt{x}}\right)}^{3}$

#### Question 9:

$\frac{2{x}^{2}+3x+4}{x}$

$\frac{d}{dx}\left(\frac{2{x}^{2}+3x+4}{x}\right)\phantom{\rule{0ex}{0ex}}=\frac{d}{dx}\left(\frac{2{x}^{2}}{x}\right)+\frac{d}{dx}\left(\frac{3x}{x}\right)+\frac{d}{dx}\left(\frac{4}{x}\right)\phantom{\rule{0ex}{0ex}}=2\frac{d}{dx}\left(x\right)+3\frac{d}{dx}\left(1\right)+4\frac{d}{dx}\left({x}^{-1}\right)\phantom{\rule{0ex}{0ex}}=2\left(1\right)+3\left(0\right)+4\left(-1\right){x}^{-2}\phantom{\rule{0ex}{0ex}}=2-\frac{4}{{x}^{2}}$

#### Question 10:

$\frac{\left({x}^{3}+1\right)\left(x-2\right)}{{x}^{2}}$

$\frac{d}{dx}\left(\frac{\left({x}^{3}+1\right)\left(x-2\right)}{{x}^{2}}\right)\phantom{\rule{0ex}{0ex}}=\frac{d}{dx}\left(\frac{{x}^{4}-2{x}^{3}+x-2}{{x}^{2}}\right)\phantom{\rule{0ex}{0ex}}=\frac{d}{dx}\left(\frac{{x}^{4}}{{x}^{2}}\right)-2\frac{d}{dx}\left(\frac{{x}^{3}}{{x}^{2}}\right)+\frac{d}{dx}\left(\frac{x}{{x}^{2}}\right)-\frac{d}{dx}\left(\frac{2}{{x}^{2}}\right)\phantom{\rule{0ex}{0ex}}=\frac{d}{dx}\left({x}^{2}\right)-2\frac{d}{dx}\left(x\right)+\frac{d}{dx}\left({x}^{-1}\right)-2\frac{d}{dx}\left({x}^{-2}\right)\phantom{\rule{0ex}{0ex}}=2x-2-\frac{1}{{x}^{2}}-2\left(-2\right){x}^{-3}\phantom{\rule{0ex}{0ex}}=2x-2-\frac{1}{{x}^{2}}+\frac{4}{{x}^{3}}$

#### Question 12:

2 sec x + 3 cot x − 4 tan x

#### Question 13:

a0 xn + a1 xn−1 + a2 xn2 + ... + an1 x + an.

#### Question 15:

$\frac{\left(x+5\right)\left(2{x}^{2}-1\right)}{x}$

$\frac{d}{dx}\left(\frac{\left(x+5\right)\left(2{x}^{2}-1\right)}{x}\right)\phantom{\rule{0ex}{0ex}}=\frac{d}{dx}\left(\frac{2{x}^{3}+10{x}^{2}-x-5}{x}\right)\phantom{\rule{0ex}{0ex}}=\frac{d}{dx}\left(\frac{2{x}^{3}}{x}\right)+\frac{d}{dx}\left(\frac{10{x}^{2}}{x}\right)-\frac{d}{dx}\left(\frac{x}{x}\right)-\frac{d}{dx}\left(\frac{5}{x}\right)\phantom{\rule{0ex}{0ex}}=2\frac{d}{dx}\left({x}^{2}\right)+10\frac{d}{dx}\left(x\right)-\frac{d}{dx}\left(1\right)-5\frac{d}{dx}\left({x}^{-1}\right)\phantom{\rule{0ex}{0ex}}=2\left(2x\right)+10\left(1\right)-0-5\left(-1\right){x}^{-2}\phantom{\rule{0ex}{0ex}}=4x+10+\frac{5}{{x}^{2}}$

cos (x + a)

#### Question 21:

Find the slope of the tangent to the curve f (x) = 2x6 + x4 − 1 at x = 1.

#### Question 23:

Find the rate at which the function f (x) = x4 − 2x3 + 3x2 + x + 5 changes with respect to x.

#### Question 25:

If for f (x) = λ x2 + μ x + 12, f' (4) = 15 and f' (2) = 11, then find λ and μ.

#### Question 26:

For the function $f\left(x\right)=\frac{{x}^{100}}{100}+\frac{{x}^{99}}{99}+...+\frac{{x}^{2}}{2}+x+1.$ Prove that f' (1) = 100 f' (0).

x3 sin x

x3 ex

x2 ex log x

xn tan x

xn loga x

#### Question 6:

(x3 + x2 + 1) sin x

sin x cos x

x2 sin x log x

x5 ex + x6 log x

#### Question 11:

(x sin x + cos x) (x cos x − sin x)

#### Question 12:

(x sin x + cos x ) (ex + x2 log x)

#### Question 13:

(1 − 2 tan x) (5 + 4 sin x)

(1 +x2) cos x

sin2 x

logx2 x

x3 ex cos x

#### Question 20:

x4 (5 sin x − 3 cos x)

(2x2 − 3) sin x

x5 (3 − 6x−9)

x4 (3 − 4x−5)

x−3 (5 + 3x)

#### Question 25:

Differentiate in two ways, using product rule and otherwise, the function (1 + 2 tan x) (5 + 4 cos x). Verify that the answers are the same.

#### Question 26:

Differentiate each of the following functions by the product rule and the other method and verify that answer from both the methods is the same.
(i) (3x2 + 2)2
(ii) (x + 2) (x + 3)
(iii) (3 sec x − 4 cosec x) (−2 sin x + 5 cos x)

#### Question 27:

(ax + b) (a + d)2

#### Question 28:

(ax + b)n (cx + d)n

#### Question 1:

$\frac{{x}^{2}+1}{x+1}$

#### Question 2:

$\frac{2x-1}{{x}^{2}+1}$

#### Question 5:

$\frac{a{x}^{2}+bx+c}{p{x}^{2}+qx+r}$

#### Question 7:

$\frac{1}{a{x}^{2}+bx+c}$

#### Question 8:

$\frac{{e}^{x}}{1+{x}^{2}}$

#### Question 14:

$\frac{{x}^{2}-x+1}{{x}^{2}+x+1}$

#### Question 15:

$\frac{\sqrt{a}+\sqrt{x}}{\sqrt{a}-\sqrt{x}}$

#### Question 16:

Let us use the quotient rule here.
We have:
u = a + sin x and v =1 + a sin x
u' = cos x and v'=a cos x

#### Question 18:

$\frac{1+{3}^{x}}{1-{3}^{x}}$

#### Question 24:

$\frac{p{x}^{2}+qx+r}{ax+b}$

#### Question 29:

$\frac{ax+b}{p{x}^{2}+qx+r}$

#### Question 30:

$\frac{1}{a{x}^{2}+bx+c}$

#### Question 1:

Write the value of $\underset{x\to c}{\mathrm{lim}}\frac{f\left(x\right)-f\left(c\right)}{x-c}$.

#### Question 2:

Write the value of .

#### Question 3:

If x < 2, then write the value of $\frac{d}{dx}\left(\sqrt{{x}^{2}-4x+4\right)}$.

#### Question 4:

If $\frac{\mathrm{\pi }}{2}$ < x < π, then find .

#### Question 5:

Write the value of .

#### Question 6:

Write the value of .

#### Question 7:

If f (x) = |x| + |x−1|, write the value of .

#### Question 8:

Write the value of the derivative of f (x) = |x − 1| + |x − 3| at x = 2.

If f (x) = .

#### Question 10:

Write the value of .

#### Question 11:

If f (1) = 1, f' (1) = 2, then write the value of .

#### Question 12:

Write the derivative of f (x) = 3 |2 + x| at x = −3.

#### Question 13:

If |x| < 1 and y = 1 + x + x2 + x3 + ..., then write the value of $\frac{dy}{dx}$.

The given series is a geometric series where a = 1 and r = x.

#### Question 14:

If f (x) = ${\mathrm{log}}_{{x}_{2}}$ x3, write the value of f' (x).

#### Question 1:

Mark the correct alternative in each of the following:

Let f(x) = x − [x], xR, then $f\text{'}\left(\frac{1}{2}\right)$ is

(a) $\frac{3}{2}$                                  (b) 1                                  (c) 0                                  (d) −1

Given: f(x) = x − [x], xR

Now,

For 0 ≤ x < 1, [x] = 0.

f(x) = x − 0 = x, ∀ x ∈ [0, 1)

Differentiating both sides with respect to x, we get

f '(x) = 1, ∀ x ∈ [0, 1)

Hence, the correct answer is option (b).

#### Question 2:

Mark the correct alternative in each of the following:

If $f\left(x\right)=\frac{x-4}{2\sqrt{x}}$, then f '(1) is

(a) $\frac{5}{4}$                                  (b) $\frac{4}{5}$                                  (c) 1                                  (d) 0

Differentiating both sides with respect to x, we get

Hence, the correct answer is option (a).

#### Question 3:

Mark the correct alternative in each of the following:

If $y=1+\frac{x}{1!}+\frac{{x}^{2}}{2!}+\frac{{x}^{3}}{3!}+...$, then $\frac{dy}{dx}=$

(a) y + 1                                 (b) y − 1                                  (c) y                                 (d) y2

$y=1+\frac{x}{1!}+\frac{{x}^{2}}{2!}+\frac{{x}^{3}}{3!}+...$

Differentiating both sides with respect to x, we get

$\therefore \frac{dy}{dx}=y$

Hence, the correct answer is option (c).

#### Question 4:

Mark the correct alternative in each of the following:

If $f\left(x\right)=1-x+{x}^{2}-{x}^{3}+...-{x}^{99}+{x}^{100}$, then $f\text{'}\left(1\right)$ equals

(a) 150                                  (b) −50                                  (c) −150                                  (d) 50

$f\left(x\right)=1-x+{x}^{2}-{x}^{3}+...-{x}^{99}+{x}^{100}$

Differentiating both sides with respect to x, we get

Putting x = 1, we get

Hence, the correct answer is option (d).

#### Question 5:

Mark the correct alternative in each of the following:

If $y=\frac{1+\frac{1}{{x}^{2}}}{1-\frac{1}{{x}^{2}}}$, then $\frac{dy}{dx}=$

(a) $-\frac{4x}{{\left({x}^{2}-1\right)}^{2}}$                                  (b) $-\frac{4x}{{x}^{2}-1}$                                  (c) $\frac{1-{x}^{2}}{4x}$                                  (d) $\frac{4x}{{x}^{2}-1}$

Differentiating both sides with respect to x, we get

Hence, the correct answer is option (a).

#### Question 6:

Mark the correct alternative in each of the following:

If $y=\sqrt{x}+\frac{1}{\sqrt{x}}$, then $\frac{dy}{dx}$ at x = 1 is

(a) 1                                  (b) $\frac{1}{2}$                                  (c) $\frac{1}{\sqrt{2}}$                                  (d) 0

Differentiating both sides with respect to x, we get

Putting x = 1, we get

${\left(\frac{dy}{dx}\right)}_{x=1}=\frac{1}{2}×1-\frac{1}{2}×1=0$

Thus, $\frac{dy}{dx}$ at x = 1 is 0.

Hence, the correct answer is option (d).

#### Question 7:

Mark the correct alternative in each of the following:

If , then $f\text{'}\left(1\right)$ is equal to

(a) 5050                                  (b) 5049                                  (c) 5051                                  (d) 50051

Differentiating both sides with respect to x, we get

Putting x = 1, we get

Hence, the correct answer is option (a).

#### Question 8:

Mark the correct alternative in each of the following:

If , then $f\text{'}\left(1\right)$ is equal to

(a) $\frac{1}{100}$                                  (b) 100                                  (c) 50                                  (d) 0

Differentiating both sides with respect to x, we get

Putting x = 1, we get

Hence, the correct answer is option (b).

#### Question 9:

Mark the correct alternative in each of the following:

If $y=\frac{\mathrm{sin}x+\mathrm{cos}x}{\mathrm{sin}x-\mathrm{cos}x}$, then $\frac{dy}{dx}$ at x = 0 is

(a) −2                                 (b) 0                                  (c) $\frac{1}{2}$                                  (d) does not exist

$y=\frac{\mathrm{sin}x+\mathrm{cos}x}{\mathrm{sin}x-\mathrm{cos}x}$

Differentiating both sides with respect to x, we get

Putting x = 0, we get

${\left(\frac{dy}{dx}\right)}_{x=0}=\frac{-2}{{\left(\mathrm{sin}0-\mathrm{cos}0\right)}^{2}}=\frac{-2}{{\left(0-1\right)}^{2}}=-2$

Thus, $\frac{dy}{dx}$ at x = 0 is −2.

Hence, the correct answer is option (a).

#### Question 10:

Mark the correct alternative in each of the following:

If $y=\frac{\mathrm{sin}\left(x+9\right)}{\mathrm{cos}x}$, then $\frac{dy}{dx}$ at x = 0 is

(a) cos 9                                  (b) sin 9                                  (c) 0                                  (d) 1

$y=\frac{\mathrm{sin}\left(x+9\right)}{\mathrm{cos}x}$

Differentiating both sides with respect to x, we get

Putting x = 0, we get

${\left(\frac{dy}{dx}\right)}_{x=0}=\frac{\mathrm{cos}9}{{\mathrm{cos}}^{2}0}=\mathrm{cos}9$                        (cos 0 = 1)

Thus, $\frac{dy}{dx}$ at x = 0 is cos 9.

Hence, the correct answer is option (a).

#### Question 11:

Mark the correct alternative in each of the following:

If $f\left(x\right)=\frac{{x}^{n}-{a}^{n}}{x-a}$, then $f\text{'}\left(a\right)$ is

(a) 1                                  (b) 0                                  (c) $\frac{1}{2}$                                  (d) does not exist

Given: $f\left(x\right)=\frac{{x}^{n}-{a}^{n}}{x-a}$

Now, f(x) is not defined at x = a. Therefore, f(x) is not differentiable at x = a.

So, $f\text{'}\left(a\right)$ does not exist.

Hence, the correct answer is option (d).

#### Question 12:

Mark the correct alternative in each of the following:

If f(x) = x sinx, then $f\text{'}\left(\frac{\mathrm{\pi }}{2}\right)=$

(a) 0                                  (b) 1                                  (c) −1                                  (d) $\frac{1}{2}$

f(x) = x sinx

Differentiating both sides with respect to x, we get

Putting $x=\frac{\mathrm{\pi }}{2}$, we get

Hence, the correct answer is option (b).

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