Chapter 2: Q. 11 (page 233)

How can the derivative of $\sin^{- 1} x$ be equal to both $\frac{1}{\sqrt{1 - x^{2}}} and \frac{1}{\cos (\sin^{- 1} x)}$ ?Which expression is easier to use, and why?

Short Answer

Expert verified

The expression $\frac{1}{\sqrt{1 - x^{2}}}$ is easier to use because it is entirely algebraic and derivative is easier to calculate at a number.

Step by step solution

Step 1. Given Information.

The given derivative of $\sin^{- 1} x$ is,

$\frac{1}{\sqrt{1 - x^{2}}} and \frac{1}{\cos (\sin^{- 1} x)}$ .

Step 2. Explanation.

Let,

$\sin^{- 1} x = θ \Rightarrow \sin θ = x N o w, \cos θ = \sqrt{1 - \sin^{2} θ} = \sqrt{1 - x^{2}} θ = \cos^{- 1} \sqrt{1 - x^{2}} \sin^{- 1} x = \cos^{- 1} \sqrt{1 - x^{2}}$

Also, we know,

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

Step 3. Easier derivative.

The derivative $\frac{1}{\sqrt{1 - x^{2}}}$ is easier to use because it is entirely algebraic and derivative is easier to calculate at a number.

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

How can the derivative of $\sin^{- 1} x$ be equal to both $\frac{1}{\sqrt{1 - x^{2}}} and \frac{1}{\cos (\sin^{- 1} x)}$ ?Which expression is easier to use, and why?

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Explanation.

Step 3. Easier derivative.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Calculus

Pure Maths

Applied Mathematics

Discrete Mathematics

Decision Maths

Study anywhere. Anytime. Across all devices.

How can the derivative of sin-1xbe equal to both11-x2and1cossin-1x?Which expression is easier to use, and why?

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Explanation.

Step 3. Easier derivative.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Calculus

Pure Maths

Applied Mathematics

Discrete Mathematics

Decision Maths

Study anywhere. Anytime. Across all devices.

How can the derivative of $\sin^{- 1} x$ be equal to both $\frac{1}{\sqrt{1 - x^{2}}} and \frac{1}{\cos (\sin^{- 1} x)}$ ?Which expression is easier to use, and why?