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.

x^{e} + e^{x} + log a.

Answer:

Let y = x^{e} + e^{x} + log a

\(\frac{d y}{d x}\) = ex^{e} + e^{x} + 0 = ex^{e} + e^{x}.

Question 2.

\(\sqrt{x+1}\)

Answer:

Let y = \(\sqrt{x+1}\)

\(\frac{d y}{d x}=\frac{1}{2 \sqrt{x+1}}\)

Question 3.

x^{e} + e^{x} + e^{e}.

Answer:

Let y = x^{e} + e^{x} + e^{e}

\(\frac{d y}{d x}\) = ex^{e-1} + e^{x} + 0

= ex^{e-1} + e^{x}

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.e^{4x}

\(\frac{d y}{d x}\) = 7.e^{4x}

= 28e^{4x}

Question 6.

\(\frac{1}{\sqrt[3]{x^{5}}}\)

Answer:

Question 7.

\(\sqrt[3]{x^{4}}\)

Answer:

Let y = (x^{4})^{1/3} = x^{4/3}

\(\frac{d y}{d x}=\frac{4}{3} x^{4 / 3-1}\)

= \(\frac{4}{3} \cdot x^{1 / 3}\)

Question 8.

e^{x2}

Answer:

Let y = e^{x2}

\(\frac{d y}{d x}\) = e^{x2} .2x.

Question 9.

Answer:

5e^{x} – logx – 3√x.

Let y = 5e^{x} – logx – 3√x

\(\frac{d y}{d x}\) = 5e^{x} – \(\frac{1}{x}-\frac{3}{2 \sqrt{x}}\)

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) .\(\frac { 1 }{ x }\)

\(=\frac{\cos (\log x)}{x}\)

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:

Question 15.

\(\sqrt{\tan x}\)

Answer:

Question 16.

cot^{3}x.

Answer:

Let y = (cot x)^{3}

\(\frac{d y}{d x}\) = 3 cot^{2} x.(-cosec^{2}x)

= – 3 cot^{2} x. cosec^{2}x.

Question 17.

(2x^{2} + 4x + 5)^{5}.

Answer:

Let y = (2x^{2} + 4x + 5)^{5}

\(\frac{d y}{d x}\) = 5(2x^{2} + 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)}

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 = 5t^{2} and y = 10t find \(\frac{d y}{d x}\).

Answer:

diff both w.r.t r

Question 2.

If x^{2} + y^{2} = a^{2} 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 x^{2} + y^{2} = 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 x^{2} + y^{2} + 2xy = 13 find \(\frac{d y}{d x}\)

Answer:

Question 5.

Differentiate log_{x2} 3 w.r.t x.

Answer:

Let y = log_{x}^{2} 3

Question 6.

If x^{2}y + y^{2} = 5 find \(\frac{d y}{d x}\).

Answer:

diff w.r.t x

Question 7.

If x = at^{2} y = 2at find \(\frac{d y}{d x}\)

Answer:

diff both w. r.t t

Question 8.

If y = x^{x} find \(\frac{d y}{d x}\)

Answer:

Taking log^{m} both sides

log y = logx^{x}

log y = x logx

diff w.t.r x

Question 9.

If y = \((\sqrt{x})^{x}\)

Answer:

Question 10.

If \(\sqrt{x}+\sqrt{y}=\sqrt{a}\) find \(\frac{d y}{d x}\) at (1,4)

Answer:

diff w.r.t x

Question 11.

If f(x) = x^{2} – 3x + 10 find f^{1}(50) and f^{1}(11).

Answer:

diff w.r.t x

f^{1}(x) = 2x – 3

f^{1}(50) = 100 – 3 = 97

f^{1}(11) = 22 – 3 = 19.

Question 12.

If f(x) = x^{n} & If f^{1} = 10 find n.

Answer:

f'(x) = n . x^{n – 1}

f^{1}(1) = n(1)^{n – 1} 1

0 =n . 1^{n – 1} ⇒ n = 10.

Question 13.

If y = log(x + 1 + x^{2}) P.T \(\frac{d y}{d x}=\frac{1}{\sqrt{1+x^{2}}}\)

Answer:

diff w.r.t x

Question 14.

If y = \(\frac{\cos x}{1+\sin x}\) find \(\frac{d y}{d x}\)

Answer:

diff w.r.t x

Question 15.

y = tan (log (sin x)) find \(\frac{d y}{d x}\)

Answer:

\(\frac{d y}{d x}\) = sec^{2}(log(sin x)). \(\frac{\cos x}{\sin x}\)

= sec^{2}(log(sin x)). cot x

Question 16.

If y = cot \(\left(x^{2}+\frac{1}{x^{2}}\right)\)

Answer:

diff w.r.t

Question 17.

If y = log\(\left(\frac{1-x^{2}}{1+x^{2}}\right)\) find \(\frac{d y}{d x}\)

Answer:

y = log (1 – x^{2}) – log (1 + x^{2})

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 Differential Calculus Three Marks Questions and Answers

Question 1.

Differentiate e^{x} by 1^{st} Principles.

Answer:

Let y = e^{x
}

Question 2.

If y = a^{x + y} S.T \(\frac{d y}{d x}=\frac{y \log a}{1-y \log a}\)

Answer:

diff w.r.t x

Question 3.

If y. e^{Y} = x S.T \(\frac{d y}{d x}=\frac{y}{x(y+1)}\)

Answer:

Question 4.

if x = y^{2} 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

Question 5.

If x^{m}. y^{n} = a^{m+n} then S.T \(\frac{d y}{d x}=\frac{-m y}{n x}\)

Answer:

Taking log^{m} bothsides

log (x^{m} . y^{n}) = (a)^{m+n}

Iogx^{m} + log y^{n} = Iog (m+n)Ìoga

diff w.r.t x

Question 6.

If \(\sqrt{x+\sqrt{x+x+}}\) ……… ∞ P.T \(\frac{d y}{d x}=\frac{1}{2 y-1}\)

Answer:

y = \(\sqrt{x+y}\) S.BS

S.B.S

y^{2} = 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) \(\frac{d y}{d x}=\frac{1}{x}\)

Answer:

y = \(\sqrt{\log x+y}\)

S.B.S

y^{2} = log x + y

diff w.r.t x

Question 8.

Differentiated a^{x} from 1^{st} Principles.

Answer:

Let y = a^{x
}

Question 9.

Differentiated x^{a} from 1^{st} Principles.

Answer:

Let y = f(x) = x^{n}

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

\(\frac{d y}{d x}\) = sec^{2}

Question 11.

If y = \((x+\sqrt{x^{2}+1})^{n}\) P.T (x^{2} + 1) \(\left(\frac{d y}{d x}\right)^{2}\) = n^{2} y^{2}

Answer:

y = \((x+\sqrt{x^{2}+1})^{n}\)

diff w.r.t

Question 12.

If e^{x} + e^{y} = e^{x+y} S.T \(\frac{d y}{d x}\) = -e^{y – x}

Answer:

diff w.r.t x

e^{x} + e^{y} \(\frac{d y}{d x}\) = e^{x+y}\(\left(1+\frac{d y}{d x}\right)\)

Question 13.

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

Answer:

Question 14.

Answer:

We have y = x^{y} apply log^{m} both sides we get

log y = y log x

Question 15.

If y = (sin x)^{tanx} find \(\frac{d y}{d x}\).

Answer:

Taking log^{m} both sides we get

log y = log (sin x)^{tan x} tan^{x}

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). sec^{2}x

\(\frac{d y}{d x}\) = y[1 + sec^{2}x. log(sin x)].

Question 16.

If x^{y} = 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 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

x^{2}(1 + y) = y^{2}(1 + x)

x^{2} + x^{2}y – y^{2} – xy^{2} = 0

(x^{2} – y^{2}) + (x^{2}y – xy^{2}) = 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 = e^{x} logx S.T xy_{2} – (2x – 1)y + (x – 1)y = 0.

Answer:

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 = e^{x} + x. \(\frac{d y}{d x}\) + y .1 ∵ e = xy_{1} – xy

xy_{2} + y_{1} – xy_{1} -xy + xy_{1} + y_{1} = o

xy_{2} + y_{1} – 2xy_{1} + xy – y = 0

xy_{2} – (2x – 1) y_{1} + (x – 1)y = 0.

Question 3.

If y = \((x+\sqrt{1+x^{2}})^{m}\) P.T (1 + x^{2})y_{2} + xy_{1} – m^{2}y = 0

Answer:

diff w.r.t x

Again diff w.r.t x

(1 + x^{2})2y_{1} y_{2} + y^{2}_{1} (2x) = m^{2}.2y.y_{1} (divide by 2y_{1})

(1 + x^{2})y_{2} + xy_{1} – m^{2}y = 0.

Question 4.

If y = \(x+\sqrt{x^{2}-1}\) S.T (x^{2} – 1)y_{2} + xy_{1} – y = 0

Answer:

diff w.r.t x

\(\sqrt{x^{2}-1}\), y_{1} = y S.B.S

(x^{2} – 1)y_{1}^{2} = y^{2}

(x^{2} – 1)2y_{1}y_{1} . 2x = 2yy_{1} + by 2y_{1} we get

(x^{2} – 1)y_{2} + xy_{1} – 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

Question 6.

Ify = (x^{2} + a^{2})^{5} S.T (x^{2} – a^{2}) \(\frac{d^{2} y}{d x^{2}}\) – 10x \(\frac{d y}{d x}\) – 12y = 0

Answer:

diff w.r.t x

\(\frac{d y}{d x}\) = 6(x^{2} + a^{2})^{5} . 2x multiply (x^{2} + a^{2})

(x^{2} + a^{2})\(\frac{d y}{d x}\) = 12(x^{2} + a^{2})

(x^{2} + a^{2})\(\frac{d y}{d x}\) = 12xy

Question 7.

If x = at^{2} & y = 2at find \(\frac{d^{2} y}{d x^{2}}\)

Answer:

diff w.r.t x

Question 8.

If y = a x^{n + 1} + \(\frac{b}{x^{n}}\) P.T. x^{2}\(\frac{d^{2} y}{d x^{2}}\) = n(n + 1)y .

Answer:

y = a.x^{n + 1} + b.x^{-n}

diff w.r.t x

y_{1} = a(n + 1)x^{n} + b – n. x^{-n-1}

y_{1} = a{n + 1)x^{n} – bn. x^{n} – 1

Again diff w.r.t x

y_{2} = a(n + 1) n.x^{n-1} – b.n(-n – 1) x^{-n-2}

y_{2} = n(n + 1) [ax^{n-1} + bx^{-n-2}] multiply both sides by x^{2}

x^{2}y_{2} = n(n + 1) (a.x^{n – 1} . x^{2} + b.x ^{-n-2}. x^{2})

= n(n + 1) a.x^{n-1} + b.x^{-n}

x^{2}y_{2} = n(n + 1 ).y

Question 9.

If 4x^{2} + 9y^{2} = 36 P.T = \(\frac{d^{2} y}{d x^{2}}=\frac{-16}{9 y^{3}}\)

Answer:

diff w.r.t x

8x + 18y.y_{1} = 0

y_{1} = \(\frac{-8 x}{18 y}=\frac{-4 x}{9 y}\) ______ (1)

Again diff w.r.t x

Question 10.

If x^{m}y^{n} = (x + y)^{m+n} S.T \(\frac{d y}{d x}=\frac{y}{x}\)

Answer:

Taking logm both sides we get

log (x^{m}.y^{n}) = 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 y^{2} + 2y = x^{2} S.T y_{1} = \(\frac{1}{(1+y)^{3}}\)

Answer:

diff w.r.t x

2yy_{1} + 2y_{1} = 2x

Question 12.

If y = 3e^{2x} + 2.e^{3x} P.T y_{2} – 5y_{1} + 6y = 0.

Answer:

diff w.r.t x

y_{1} = 6e^{2x} + 6.e^{3x}

y_{1} = 6(e^{2x} + e^{3x})

y_{2} = 6(2e^{2x} + 3e^{3x})

Consider LH.S = y_{2} – 5y_{1} + 6y

= 0 = R.H.S.

Question 13.

If y = a cos (log x) + b sin(logx) S.T x^{2}y_{2} + xy_{1} +y = 0.

Answer:

diff w.r.t x

y_{1} = a\(\left[\frac{-\sin (\log x)}{x}\right]+b\left[\frac{\cos (\log x)}{x}\right]\)

xy_{1} = -a sin (Iogx) + b cos(1ogx)

diff again w.r.t x

xy_{2} + y_{1} = \(-\frac{a \cos (\log x)}{x}-\frac{b \sin (\log x)}{x}\)

x^{2}y_{2} + xy_{1} = -(a cos log x + b sin(1ogx)

x^{2}y_{2} + x_{1} + y = 0.

Question 14.

If y = sin (logx) S.T x^{2}y_{2} + xy_{1} + y = 0.

Answer:

diff w.r.t x

y_{1} = \(\frac{\cos (\log x)}{x}\)

xy_{1} = cos(logx)

duff again w.r.t x

x.y^{2} + y^{1} . 1 = \(\frac{-\sin (\log x)}{x}\)

⇒ x^{2}y_{2} + xy_{1} = 0

Question 15.

Differentiate cotx from I^{st} Principles.

Answer:

Let y = f(x) = cot t

Question 16.

If y = x^{logx} + (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 = x^{logx} and V = (log x)^{x}

Taking log^{m} both sides

log u = log x. log x,

log u = (log x)^{2}

diff both w.r.tx logV = x . log (log x)

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