Odisha State Board Elements of Mathematics Class 12 CHSE Odisha Solutions Chapter 7 Continuity and Differentiability Ex 7(d) Textbook Exercise Questions and Answers.

## CHSE Odisha Class 12 Math Solutions Chapter 7 Continuity and Differentiability Exercise 7(d)

Question 1.

Prove the formulae (4) to (7).

Solution:

(4) \(\frac{d}{d x}\)(cos-1 x) = \(\frac{-1}{\sqrt{1-x^2}}\)

Let y = cos^{-1} x

⇒ x = cos y

⇒ \(\frac{d}{d x}\) = \(\frac{1}{\left(\frac{d x}{d y}\right)}\) = \(\frac{1}{-\sin y}\)

But sin y ≥ 0 when

y ∈ [0, π] ( ∵ [0, π] is the principal value branch for cos^{-1} x)

∴ \(\frac{d y}{d x}\) = \(\frac{-1}{\sqrt{1-\cos ^2 y}}\) = \(\frac{-1}{\sqrt{1-x^2}}\)

(5) Let y = tan^{-1} x

⇒ x = tan y

(6) Let y = cot^{-1} x

⇒ x = cot y

(7) Let y = cosec^{-1} x

⇒ x = cosec y

Question 2.

Find derivatives of the following functions.

sin^{-1} 2x

Solution:

y = sin^{-1} 2x

Question 3.

cot^{-1} √x

Solution:

cot^{-1} √x

Question 4.

sec^{-1} (2x + 1)

Solution:

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

Question 5.

cos^{-1} \(\sqrt{\frac{1+x}{2}}\)

Solution:

Question 6.

cos^{-1} \(\left(\frac{x-\frac{1}{x}}{x+\frac{1}{x}}\right)\)

Solution:

Question 7.

tan^{-1} (cos √x)

Solution:

y = tan^{-1} (cos √x)

Question 8.

x^{2} cosec^{-1} \(\left(\frac{1}{\ln x}\right)\)

Solution:

Question 9.

cot^{-1} \( \frac{\sqrt{1-x^2}}{x} \)

Solution:

Question 10.

(x sin^{-1} x)^{15}

Solution:

y = (x sin^{-1} x)^{15}

Question 11.

sin^{-1} \( \sqrt{\frac{1-x}{1+x}} \)

Solution: