2nd PUC Basic Maths Question Bank Chapter 18 Differential Calculus

Students can Download Basic Maths Question Bank Chapter 18 Differential Calculus Questions and Answers, Notes Pdf, 2nd PUC Basic Maths Question Bank with Answers helps you to revise the complete Karnataka State Board Syllabus and to clear all their doubts, score well in final exams.

Karnataka 2nd PUC Basic Maths Question Bank Chapter 18 Differential Calculus

2nd PUC Basic Maths Differential Calculus One Mark Questions and Answers

Question 1.
xe + ex + log a.
Answer:
Let y = xe + ex + log a
\(\frac{d y}{d x}\) = exe + ex + 0 = exe + ex.

Question 2.
\(\sqrt{x+1}\)
Answer:
Let y = \(\sqrt{x+1}\)
\(\frac{d y}{d x}=\frac{1}{2 \sqrt{x+1}}\)

2nd PUC Basic Maths Question Bank Chapter 18 Differential Calculus

Question 3.
xe + ex + ee.
Answer:
Let y = xe + ex + ee
\(\frac{d y}{d x}\) = exe-1 + ex + 0
= exe-1 + ex

Question 4.
log (3x + 5).
Answer;
Let log (3x + 5)
\(\frac{d y}{d x}=\frac{3}{3 x+5}\)

Question 5.
\(\frac{7}{e^{-4 x}}\)
Answer:
Let y = 7.e4x
\(\frac{d y}{d x}\) = 7.e4x
= 28e4x

Question 6.
\(\frac{1}{\sqrt[3]{x^{5}}}\)
Answer:
2nd PUC Basic Maths Question Bank Chapter 18 Differential Calculus 1

2nd PUC Basic Maths Question Bank Chapter 18 Differential Calculus

Question 7.
\(\sqrt[3]{x^{4}}\)
Answer:
Let y = (x4)1/3 = x4/3
\(\frac{d y}{d x}=\frac{4}{3} x^{4 / 3-1}\)
= \(\frac{4}{3} \cdot x^{1 / 3}\)

Question 8.
ex2
Answer:
Let y = ex2
\(\frac{d y}{d x}\) = ex2 .2x.

Question 9.
Answer:
5ex – logx – 3√x.
Let y = 5ex – logx – 3√x
\(\frac{d y}{d x}\) = 5ex – \(\frac{1}{x}-\frac{3}{2 \sqrt{x}}\)

Question 10.
log (cos x)
Answer:
Let y = log (cos x)
\(\frac{d y}{d x}\) = \(\frac{1}{\cos x}\) . – sin x = -tanx

Question 11.
sin (log x)
Answer:
Let y = sin (log x)
\(\frac{d y}{d x}\) = cos(log x) .\(\frac { 1 }{ x }\)
\(=\frac{\cos (\log x)}{x}\)

2nd PUC Basic Maths Question Bank Chapter 18 Differential Calculus

Question 12.
log (Iogx).
Answer:
Let y = log (log x)
\(\frac{d y}{d x}=\frac{1}{x \cdot \log x}\)

Question 13.
\(e^{\sqrt{x}}\)
Answer:
Let y = \(e^{\sqrt{x}}\)
\(\frac{d y}{d x}=e^{\sqrt{x}} \cdot \frac{1}{2 \sqrt{x}}\)

Question 14.
\(\sqrt{4 x+7}\)
Answer:
2nd PUC Basic Maths Question Bank Chapter 18 Differential Calculus 2

Question 15.
\(\sqrt{\tan x}\)
Answer:
2nd PUC Basic Maths Question Bank Chapter 18 Differential Calculus 3

2nd PUC Basic Maths Question Bank Chapter 18 Differential Calculus

Question 16.
cot3x.
Answer:
Let y = (cot x)3
\(\frac{d y}{d x}\) = 3 cot2 x.(-cosec2x)
= – 3 cot2 x. cosec2x.

Question 17.
(2x2 + 4x + 5)5.
Answer:
Let y = (2x2 + 4x + 5)5
\(\frac{d y}{d x}\) = 5(2x2 + 4x + 5)4(4x + 4).

Question 18.
tan √x.
Answer:
Let y = tan √x
\(\frac{d y}{d x}=\frac{\sec ^{2} \sqrt{x}}{2 \sqrt{x}}\)

Question 19.
log (ax + b)
Answer:
Let y = log (ax + b)
\(\frac{d y}{d x}=\frac{a}{a x+b}\)

Question 20.
e(5x + 6)
Answer:
Let y = e(5x + 6)
\(\frac{d y}{d x}\) = 5 . e(5x + 6)

2nd PUC Basic Maths Question Bank Chapter 18 Differential Calculus

Question 21.
log (6 – 5x)
Answer:
Let y = log (6 – 5x)
\(\frac{d y}{d x}=\frac{-5}{6-5 x}\)

Question 22.
e-3x2
Answer
Let y = e-3x2
\(\frac{d y}{d x}\) = -e-3x2 .6x.

2nd PUC Basic Maths Differential Calculus Two Marks Questions and Answers

Question 1.
If x = 5t2 and y = 10t find \(\frac{d y}{d x}\).
Answer:
diff both w.r.t r
2nd PUC Basic Maths Question Bank Chapter 18 Differential Calculus 4

Question 2.
If x2 + y2 = a2 find
Answer:
diff w.r.t x
2x + 2y \(\frac{d y}{d x}\) = 0 ⇒ \(\frac{d y}{d x}=-\frac{x}{y}\)

Question 3.
If x2 + y2 = 13 find dy/dx when x = 3 and y = -2.
Answer:
diff w.r.t x
2x + 2y \(\frac{d y}{d x}\) = 0
\(\frac{d y}{d x}=-\frac{-2 x}{2 y}=\frac{-3}{-2}=\frac{3}{2}\)

Question 4.
If x2 + y2 + 2xy = 13 find \(\frac{d y}{d x}\)
Answer:
2nd PUC Basic Maths Question Bank Chapter 18 Differential Calculus 5

Question 5.
Differentiate logx2 3 w.r.t x.
Answer:
Let y = logx2 3
2nd PUC Basic Maths Question Bank Chapter 18 Differential Calculus 6

2nd PUC Basic Maths Question Bank Chapter 18 Differential Calculus

Question 6.
If x2y + y2 = 5 find \(\frac{d y}{d x}\).
Answer:
diff w.r.t x
2nd PUC Basic Maths Question Bank Chapter 18 Differential Calculus 7

Question 7.
If x = at2 y = 2at find \(\frac{d y}{d x}\)
Answer:
diff both w. r.t t
2nd PUC Basic Maths Question Bank Chapter 18 Differential Calculus 8

Question 8.
If y = xx find \(\frac{d y}{d x}\)
Answer:
Taking logm both sides
log y = logxx
log y = x logx
diff w.t.r x
2nd PUC Basic Maths Question Bank Chapter 18 Differential Calculus 9

2nd PUC Basic Maths Question Bank Chapter 18 Differential Calculus

Question 9.
If y = \((\sqrt{x})^{x}\)
Answer:
2nd PUC Basic Maths Question Bank Chapter 18 Differential Calculus 10

Question 10.
If \(\sqrt{x}+\sqrt{y}=\sqrt{a}\) find \(\frac{d y}{d x}\) at (1,4)
Answer:
diff w.r.t x
2nd PUC Basic Maths Question Bank Chapter 18 Differential Calculus 11

Question 11.
If f(x) = x2 – 3x + 10 find f1(50) and f1(11).
Answer:
diff w.r.t x
f1(x) = 2x – 3
f1(50) = 100 – 3 = 97
f1(11) = 22 – 3 = 19.

Question 12.
If f(x) = xn & If f1 = 10 find n.
Answer:
f'(x) = n . xn – 1
f1(1) = n(1)n – 1 1
0 =n . 1n – 1 ⇒ n = 10.

2nd PUC Basic Maths Question Bank Chapter 18 Differential Calculus

Question 13.
If y = log(x + 1 + x2) P.T \(\frac{d y}{d x}=\frac{1}{\sqrt{1+x^{2}}}\)
Answer:
diff w.r.t x
2nd PUC Basic Maths Question Bank Chapter 18 Differential Calculus 12

Question 14.
If y = \(\frac{\cos x}{1+\sin x}\) find \(\frac{d y}{d x}\)
Answer:
diff w.r.t x
2nd PUC Basic Maths Question Bank Chapter 18 Differential Calculus 13

2nd PUC Basic Maths Question Bank Chapter 18 Differential Calculus

Question 15.
y = tan (log (sin x)) find \(\frac{d y}{d x}\)
Answer:
\(\frac{d y}{d x}\) = sec2(log(sin x)). \(\frac{\cos x}{\sin x}\)
= sec2(log(sin x)). cot x

Question 16.
If y = cot \(\left(x^{2}+\frac{1}{x^{2}}\right)\)
Answer:
diff w.r.t
2nd PUC Basic Maths Question Bank Chapter 18 Differential Calculus 14

Question 17.
If y = log\(\left(\frac{1-x^{2}}{1+x^{2}}\right)\) find \(\frac{d y}{d x}\)
Answer:
y = log (1 – x2) – log (1 + x2)
2nd PUC Basic Maths Question Bank Chapter 18 Differential Calculus 15

Question 18.
y = log\(\left(\frac{1+\sin x}{1-\sin x}\right)\) find \(\frac{d y}{d x}\)
Answer:
y = log(1 + sinx) – log (1 – sin x)
2nd PUC Basic Maths Question Bank Chapter 18 Differential Calculus 16

2nd PUC Basic Maths Question Bank Chapter 18 Differential Calculus

2nd PUC Basic Maths Differential Calculus Three Marks Questions and Answers

Question 1.
Differentiate ex by 1st Principles.
Answer:
Let y = ex
2nd PUC Basic Maths Question Bank Chapter 18 Differential Calculus 17

2nd PUC Basic Maths Question Bank Chapter 18 Differential Calculus

Question 2.
If y = ax + y S.T \(\frac{d y}{d x}=\frac{y \log a}{1-y \log a}\)
Answer:
diff w.r.t x
2nd PUC Basic Maths Question Bank Chapter 18 Differential Calculus 18

Question 3.
If y. eY = x S.T \(\frac{d y}{d x}=\frac{y}{x(y+1)}\)
Answer:
2nd PUC Basic Maths Question Bank Chapter 18 Differential Calculus 19

Question 4.
if x = y2 log x S.T \(\frac{d y}{d x}=\frac{y\left(x-y^{2}\right)}{2 x^{2}}\)
Answer:
diff w.r.t x
2nd PUC Basic Maths Question Bank Chapter 18 Differential Calculus 20
2nd PUC Basic Maths Question Bank Chapter 18 Differential Calculus 48

2nd PUC Basic Maths Question Bank Chapter 18 Differential Calculus

Question 5.
If xm. yn = am+n then S.T \(\frac{d y}{d x}=\frac{-m y}{n x}\)
Answer:
Taking logm bothsides
log (xm . yn) = (a)m+n
Iogxm + log yn = Iog (m+n)Ìoga
diff w.r.t x
2nd PUC Basic Maths Question Bank Chapter 18 Differential Calculus 21

Question 6.
If \(\sqrt{x+\sqrt{x+x+}}\) ……… ∞ P.T \(\frac{d y}{d x}=\frac{1}{2 y-1}\)
Answer:
y = \(\sqrt{x+y}\) S.BS
S.B.S
y2 = x + y
diff w.r.t x
2nd PUC Basic Maths Question Bank Chapter 18 Differential Calculus - 23

Question 7.
If y = \(\sqrt{\log x+\sqrt{\log x+\log x \ldots . . \infty}}\) S.T (2y – 1) \(\frac{d y}{d x}=\frac{1}{x}\)
Answer:
y = \(\sqrt{\log x+y}\)
S.B.S
y2 = log x + y
diff w.r.t x

2nd PUC Basic Maths Question Bank Chapter 18 Differential Calculus 22

2nd PUC Basic Maths Question Bank Chapter 18 Differential Calculus

Question 8.
Differentiated ax from 1st Principles.
Answer:
Let y = ax

2nd PUC Basic Maths Question Bank Chapter 18 Differential Calculus 24

Question 9.
Differentiated xa from 1st Principles.
Answer:
Let y = f(x) = xn

2nd PUC Basic Maths Question Bank Chapter 18 Differential Calculus 25

2nd PUC Basic Maths Question Bank Chapter 18 Differential Calculus

Question 10.
If y = \(\sqrt{\frac{1-\cos 2 x}{1+\cos 2 x}}\) find \(\frac{d y}{d x}\)
Answer:
y = \(\sqrt{\frac{2 \sin ^{2} x}{2 \cos ^{2} x}}\)
y = tan x
\(\frac{d y}{d x}\) = sec2

Question 11.
If y = \((x+\sqrt{x^{2}+1})^{n}\) P.T (x2 + 1) \(\left(\frac{d y}{d x}\right)^{2}\) = n2 y2
Answer:
y = \((x+\sqrt{x^{2}+1})^{n}\)
diff w.r.t
2nd PUC Basic Maths Question Bank Chapter 18 Differential Calculus 26

Question 12.
If ex + ey = ex+y S.T \(\frac{d y}{d x}\) = -ey – x
Answer:
diff w.r.t x
ex + ey \(\frac{d y}{d x}\) = ex+y\(\left(1+\frac{d y}{d x}\right)\)
2nd PUC Basic Maths Question Bank Chapter 18 Differential Calculus 27

2nd PUC Basic Maths Question Bank Chapter 18 Differential Calculus

Question 13.
If sin y = x sin (a +y) then P.T \(\frac{d y}{d x}=\frac{\sin ^{2}(a+y)}{\sin a}\)
Answer:
2nd PUC Basic Maths Question Bank Chapter 18 Differential Calculus 28

Question 14.
2nd PUC Basic Maths Question Bank Chapter 18 Differential Calculus 29
Answer:
We have y = xy apply logm both sides we get
log y = y log x
2nd PUC Basic Maths Question Bank Chapter 18 Differential Calculus 30

2nd PUC Basic Maths Question Bank Chapter 18 Differential Calculus

Question 15.
If y = (sin x)tanx find \(\frac{d y}{d x}\).
Answer:
Taking logm both sides we get
log y = log (sin x)tan x tanx
log y = tan x. log (sin x)
\(\frac{1}{y} \cdot \frac{d y}{d x}=\) = tan x . \(\frac{\cos x}{\sin x}\) + log (sin x). sec2x
\(\frac{d y}{d x}\) = y[1 + sec2x. log(sin x)].

Question 16.
If xy = P.T \(\frac{d y}{d x}=\frac{\log x}{(1+\log x)^{2}}\)
Answer:
Taking log m both sides
y log x = (x – y)
= x – y ∵ log e = 1
y + y log x = x
y(1 + logx) = x
2nd PUC Basic Maths Question Bank Chapter 18 Differential Calculus 31

2nd PUC Basic Maths Differential Calculus Five Marks Questions and Answers

Question 1.
If \(x \sqrt{1+y}+y \sqrt{1+x}=0\) = 0 where x ≠ y S.T \(\frac{d y}{d x}=\frac{-1}{(1+x)^{2}}\)
Answer:
We have \(x \sqrt{1+y}=-y \sqrt{1+x}\) S.B.S we get
x2(1 + y) = y2(1 + x)
x2 + x2y – y2 – xy2 = 0
(x2 – y2) + (x2y – xy2) = 0
(x – y) (x + y) + xy (x – y) = 0
(x – y)(x + y + xy) = 0 ⇒
x + y + xy = 0 ∵ x ≠ y
diff w.r.t t y(1 + x) = -x ⇒ y = \(\frac{-x}{1+x}\)
2nd PUC Basic Maths Question Bank Chapter 18 Differential Calculus 32

2nd PUC Basic Maths Question Bank Chapter 18 Differential Calculus

Question 2.
If y = ex logx S.T xy2 – (2x – 1)y + (x – 1)y = 0.
Answer:
diff w.r.t x
2nd PUC Basic Maths Question Bank Chapter 18 Differential Calculus 33
diff again w.r.t x d2y dy
x \(\frac{d^{2} y}{d x^{2}}+\frac{d y}{d x}\) . 1 = ex + x. \(\frac{d y}{d x}\) + y .1 ∵ e = xy1 – xy
xy2 + y1 – xy1 -xy + xy1 + y1 = o
xy2 + y1 – 2xy1 + xy – y = 0
xy2 – (2x – 1) y1 + (x – 1)y = 0.

Question 3.
If y = \((x+\sqrt{1+x^{2}})^{m}\) P.T (1 + x2)y2 + xy1 – m2y = 0
Answer:
diff w.r.t x
2nd PUC Basic Maths Question Bank Chapter 18 Differential Calculus 34
Again diff w.r.t x
(1 + x2)2y1 y2 + y21 (2x) = m2.2y.y1 (divide by 2y1)
(1 + x2)y2 + xy1 – m2y = 0.

2nd PUC Basic Maths Question Bank Chapter 18 Differential Calculus

Question 4.
If y = \(x+\sqrt{x^{2}-1}\) S.T (x2 – 1)y2 + xy1 – y = 0
Answer:
diff w.r.t x
2nd PUC Basic Maths Question Bank Chapter 18 Differential Calculus 35
\(\sqrt{x^{2}-1}\), y1 = y S.B.S
(x2 – 1)y12 = y2
(x2 – 1)2y1y1 . 2x = 2yy1 + by 2y1 we get
(x2 – 1)y2 + xy1 – y = 0.

Question 5.
If xy + 4y = 3x S.T \(\frac{d^{2} y}{d x^{2}}=\frac{-2 y}{(x+4)^{3}}\)
Answer:
y(x + 4) = 3x
2nd PUC Basic Maths Question Bank Chapter 18 Differential Calculus 36

Question 6.
Ify = (x2 + a2)5 S.T (x2 – a2) \(\frac{d^{2} y}{d x^{2}}\) – 10x \(\frac{d y}{d x}\) – 12y = 0
Answer:
diff w.r.t x
\(\frac{d y}{d x}\) = 6(x2 + a2)5 . 2x multiply (x2 + a2)
(x2 + a2)\(\frac{d y}{d x}\) = 12(x2 + a2)
(x2 + a2)\(\frac{d y}{d x}\) = 12xy
2nd PUC Basic Maths Question Bank Chapter 18 Differential Calculus 37

2nd PUC Basic Maths Question Bank Chapter 18 Differential Calculus

Question 7.
If x = at2 & y = 2at find \(\frac{d^{2} y}{d x^{2}}\)
Answer:
diff w.r.t x
2nd PUC Basic Maths Question Bank Chapter 18 Differential Calculus 38

Question 8.
If y = a xn + 1 + \(\frac{b}{x^{n}}\) P.T. x2\(\frac{d^{2} y}{d x^{2}}\) = n(n + 1)y .
Answer:
y = a.xn + 1 + b.x-n
diff w.r.t x
y1 = a(n + 1)xn + b – n. x-n-1
y1 = a{n + 1)xn – bn. xn – 1
Again diff w.r.t x
y2 = a(n + 1) n.xn-1 – b.n(-n – 1) x-n-2
y2 = n(n + 1) [axn-1 + bx-n-2] multiply both sides by x2
x2y2 = n(n + 1) (a.xn – 1 . x2 + b.x -n-2. x2)
= n(n + 1) a.xn-1 + b.x-n
x2y2 = n(n + 1 ).y

2nd PUC Basic Maths Question Bank Chapter 18 Differential Calculus

Question 9.
If 4x2 + 9y2 = 36 P.T = \(\frac{d^{2} y}{d x^{2}}=\frac{-16}{9 y^{3}}\)
Answer:
diff w.r.t x
8x + 18y.y1 = 0
y1 = \(\frac{-8 x}{18 y}=\frac{-4 x}{9 y}\) ______ (1)
Again diff w.r.t x
2nd PUC Basic Maths Question Bank Chapter 18 Differential Calculus 39

Question 10.
If xmyn = (x + y)m+n S.T \(\frac{d y}{d x}=\frac{y}{x}\)
Answer:
Taking logm both sides we get
log (xm.yn) = log (x + y)m+n</sup
m log x + n log y = (m + n) log (x + y)
diff w.r.t x
2nd PUC Basic Maths Question Bank Chapter 18 Differential Calculus 40
2nd PUC Basic Maths Question Bank Chapter 18 Differential Calculus 41

Question 11.
If y2 + 2y = x2 S.T y1 = \(\frac{1}{(1+y)^{3}}\)
Answer:
diff w.r.t x
2yy1 + 2y1 = 2x
2nd PUC Basic Maths Question Bank Chapter 18 Differential Calculus 42

2nd PUC Basic Maths Question Bank Chapter 18 Differential Calculus

Question 12.
If y = 3e2x + 2.e3x P.T y2 – 5y1 + 6y = 0.
Answer:
diff w.r.t x
y1 = 6e2x + 6.e3x
y1 = 6(e2x + e3x)
y2 = 6(2e2x + 3e3x)
Consider LH.S = y2 – 5y1 + 6y
2nd PUC Basic Maths Question Bank Chapter 18 Differential Calculus 43
= 0 = R.H.S.

Question 13.
If y = a cos (log x) + b sin(logx) S.T x2y2 + xy1 +y = 0.
Answer:
diff w.r.t x
y1 = a\(\left[\frac{-\sin (\log x)}{x}\right]+b\left[\frac{\cos (\log x)}{x}\right]\)
xy1 = -a sin (Iogx) + b cos(1ogx)
diff again w.r.t x
xy2 + y1 = \(-\frac{a \cos (\log x)}{x}-\frac{b \sin (\log x)}{x}\)
x2y2 + xy1 = -(a cos log x + b sin(1ogx)
x2y2 + x1 + y = 0.

Question 14.
If y = sin (logx) S.T x2y2 + xy1 + y = 0.
Answer:
diff w.r.t x
y1 = \(\frac{\cos (\log x)}{x}\)
xy1 = cos(logx)
duff again w.r.t x
x.y2 + y1 . 1 = \(\frac{-\sin (\log x)}{x}\)
⇒ x2y2 + xy1 = 0

2nd PUC Basic Maths Question Bank Chapter 18 Differential Calculus

Question 15.
Differentiate cotx from Ist Principles.
Answer:
Let y = f(x) = cot t
2nd PUC Basic Maths Question Bank Chapter 18 Differential Calculus 44

2nd PUC Basic Maths Question Bank Chapter 18 Differential Calculus 45

Question 16.
If y = xlogx + (log x)x find \(\frac{d y}{d x}\)
Answer:
Let y = u + v
\(\frac{d y}{d x}=\frac{d u}{d x}+\frac{d v}{d x}\) ______ (1)
where u = xlogx and V = (log x)x
Taking logm both sides
log u = log x. log x,
log u = (log x)2
diff both w.r.tx logV = x . log (log x)
2nd PUC Basic Maths Question Bank Chapter 18 Differential Calculus 46

2nd PUC Basic Maths Question Bank Chapter 18 Differential Calculus

Question 17.
If x = a[cos t + log tan \(\frac { t }{ 2 }\)] y = a sin t, S.T \(\frac{d y}{d x}\) = tan t.
Answer:
diff w.r.t x
2nd PUC Basic Maths Question Bank Chapter 18 Differential Calculus 47

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