Q. 11 How can the derivative of sin-1x... [FREE SOLUTION]

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,

$\sin^{- 1} x = 胃鈬� \sin 胃 = x N o w, \cos 胃 = \sqrt{1 - \sin^{2} 胃} = \sqrt{1 - x^{2}} 胃 = \cos^{- 1} \sqrt{1 - x^{2}} \sin^{- 1} x = \cos^{- 1} \sqrt{1 - x^{2}}$

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.

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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.

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91影视

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

Applied Mathematics

Calculus

Geometry

Logic and Functions

Decision Maths

Theoretical and Mathematical Physics

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?