# 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.
Let y = xe + ex + log a
$$\frac{d y}{d x}$$ = exe + ex + 0 = exe + ex.

Question 2.
$$\sqrt{x+1}$$
Let y = $$\sqrt{x+1}$$
$$\frac{d y}{d x}=\frac{1}{2 \sqrt{x+1}}$$

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

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

Question 5.
$$\frac{7}{e^{-4 x}}$$
Let y = 7.e4x
$$\frac{d y}{d x}$$ = 7.e4x
= 28e4x

Question 6.
$$\frac{1}{\sqrt[3]{x^{5}}}$$

Question 7.
$$\sqrt[3]{x^{4}}$$
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
Let y = ex2
$$\frac{d y}{d x}$$ = ex2 .2x.

Question 9.
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)
Let y = log (cos x)
$$\frac{d y}{d x}$$ = $$\frac{1}{\cos x}$$ . – sin x = -tanx

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

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

Question 13.
$$e^{\sqrt{x}}$$
Let y = $$e^{\sqrt{x}}$$
$$\frac{d y}{d x}=e^{\sqrt{x}} \cdot \frac{1}{2 \sqrt{x}}$$

Question 14.
$$\sqrt{4 x+7}$$

Question 15.
$$\sqrt{\tan x}$$

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

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

Question 18.
tan √x.
Let y = tan √x
$$\frac{d y}{d x}=\frac{\sec ^{2} \sqrt{x}}{2 \sqrt{x}}$$

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

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

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

Question 22.
e-3x2
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 $$\frac{d y}{d x}$$.
diff both w.r.t r

Question 2.
If x2 + y2 = a2 find
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.
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 $$\frac{d y}{d x}$$

Question 5.
Differentiate logx2 3 w.r.t x.
Let y = logx2 3

Question 6.
If x2y + y2 = 5 find $$\frac{d y}{d x}$$.
diff w.r.t x

Question 7.
If x = at2 y = 2at find $$\frac{d y}{d x}$$
diff both w. r.t t

Question 8.
If y = xx find $$\frac{d y}{d x}$$
Taking logm both sides
log y = logxx
log y = x logx
diff w.t.r x

Question 9.
If y = $$(\sqrt{x})^{x}$$

Question 10.
If $$\sqrt{x}+\sqrt{y}=\sqrt{a}$$ find $$\frac{d y}{d x}$$ at (1,4)
diff w.r.t x

Question 11.
If f(x) = x2 – 3x + 10 find f1(50) and f1(11).
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.
f'(x) = n . xn – 1
f1(1) = n(1)n – 1 1
0 =n . 1n – 1 ⇒ n = 10.

Question 13.
If y = log(x + 1 + x2) P.T $$\frac{d y}{d x}=\frac{1}{\sqrt{1+x^{2}}}$$
diff w.r.t x

Question 14.
If y = $$\frac{\cos x}{1+\sin x}$$ find $$\frac{d y}{d x}$$
diff w.r.t x

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

Question 16.
If y = cot $$\left(x^{2}+\frac{1}{x^{2}}\right)$$
diff w.r.t

Question 17.
If y = log$$\left(\frac{1-x^{2}}{1+x^{2}}\right)$$ find $$\frac{d y}{d x}$$
y = log (1 – x2) – log (1 + x2)

Question 18.
y = log$$\left(\frac{1+\sin x}{1-\sin x}\right)$$ find $$\frac{d y}{d x}$$
y = log(1 + sinx) – log (1 – sin x)

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

Question 1.
Differentiate ex by 1st Principles.
Let y = ex

Question 2.
If y = ax + y S.T $$\frac{d y}{d x}=\frac{y \log a}{1-y \log a}$$
diff w.r.t x

Question 3.
If y. eY = x S.T $$\frac{d y}{d x}=\frac{y}{x(y+1)}$$

Question 4.
if x = y2 log x S.T $$\frac{d y}{d x}=\frac{y\left(x-y^{2}\right)}{2 x^{2}}$$
diff w.r.t x

Question 5.
If xm. yn = am+n then S.T $$\frac{d y}{d x}=\frac{-m y}{n x}$$
Taking logm bothsides
log (xm . yn) = (a)m+n
Iogxm + log yn = Iog (m+n)Ìoga
diff w.r.t x

Question 6.
If $$\sqrt{x+\sqrt{x+x+}}$$ ……… ∞ P.T $$\frac{d y}{d x}=\frac{1}{2 y-1}$$
y = $$\sqrt{x+y}$$ S.BS
S.B.S
y2 = x + y
diff w.r.t x

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

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

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

Question 10.
If y = $$\sqrt{\frac{1-\cos 2 x}{1+\cos 2 x}}$$ find $$\frac{d y}{d x}$$
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
y = $$(x+\sqrt{x^{2}+1})^{n}$$
diff w.r.t

Question 12.
If ex + ey = ex+y S.T $$\frac{d y}{d x}$$ = -ey – x
diff w.r.t x
ex + ey $$\frac{d y}{d x}$$ = ex+y$$\left(1+\frac{d y}{d x}\right)$$

Question 13.
If sin y = x sin (a +y) then P.T $$\frac{d y}{d x}=\frac{\sin ^{2}(a+y)}{\sin a}$$

Question 14.

We have y = xy apply logm both sides we get
log y = y log x

Question 15.
If y = (sin x)tanx find $$\frac{d y}{d x}$$.
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 $$\frac{d y}{d x}=\frac{\log x}{(1+\log x)^{2}}$$
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 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 $$\frac{d y}{d x}=\frac{-1}{(1+x)^{2}}$$
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 = $$\frac{-x}{1+x}$$

Question 2.
If y = ex logx S.T xy2 – (2x – 1)y + (x – 1)y = 0.
diff w.r.t x

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
diff w.r.t x

Again diff w.r.t x
(1 + x2)2y1 y2 + y21 (2x) = m2.2y.y1 (divide by 2y1)
(1 + x2)y2 + xy1 – m2y = 0.

Question 4.
If y = $$x+\sqrt{x^{2}-1}$$ S.T (x2 – 1)y2 + xy1 – y = 0
diff w.r.t x

$$\sqrt{x^{2}-1}$$, 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 $$\frac{d^{2} y}{d x^{2}}=\frac{-2 y}{(x+4)^{3}}$$
y(x + 4) = 3x

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
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

Question 7.
If x = at2 & y = 2at find $$\frac{d^{2} y}{d x^{2}}$$
diff w.r.t x

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 .
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

Question 9.
If 4x2 + 9y2 = 36 P.T = $$\frac{d^{2} y}{d x^{2}}=\frac{-16}{9 y^{3}}$$
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

Question 10.
If xmyn = (x + y)m+n S.T $$\frac{d y}{d x}=\frac{y}{x}$$
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

Question 11.
If y2 + 2y = x2 S.T y1 = $$\frac{1}{(1+y)^{3}}$$
diff w.r.t x
2yy1 + 2y1 = 2x

Question 12.
If y = 3e2x + 2.e3x P.T y2 – 5y1 + 6y = 0.
diff w.r.t x
y1 = 6e2x + 6.e3x
y1 = 6(e2x + e3x)
y2 = 6(2e2x + 3e3x)
Consider LH.S = y2 – 5y1 + 6y

= 0 = R.H.S.

Question 13.
If y = a cos (log x) + b sin(logx) S.T x2y2 + xy1 +y = 0.
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.
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

Question 15.
Differentiate cotx from Ist Principles.
Let y = f(x) = cot t

Question 16.
If y = xlogx + (log x)x find $$\frac{d y}{d x}$$
$$\frac{d y}{d x}=\frac{d u}{d x}+\frac{d v}{d x}$$ ______ (1)
If x = a[cos t + log tan $$\frac { t }{ 2 }$$] y = a sin t, S.T $$\frac{d y}{d x}$$ = tan t.