Question Statement

Find the derivative of the function:

$y = sin^{- 1} (5 x - 1)$

Background and Explanation

This problem involves differentiating an inverse trigonometric function using the chain rule. You should be familiar with the standard derivative formula for $sin^{- 1} (x)$ and how to handle composite functions where the argument is not just $x$ but a linear expression $(5 x - 1)$ .

Solution

We are given the function:

$y = sin^{- 1} (5 x - 1)$

To differentiate this with respect to $x$ , we recognize this as a composite function where:

The outer function is $sin^{- 1} (u)$
The inner function is $u = 5 x - 1$

Step 1: Recall the derivative formula for inverse sine: $\frac{d}{d u} [sin^{- 1} (u)] = \frac{1}{1 - u ^{2}}$

Step 2: Apply the chain rule, which states $\frac{d y}{d x} = \frac{d y}{d u} \cdot \frac{d u}{d x}$ :

$\frac{d y}{d x} = \frac{1}{1 - ( 5 x - 1 ) ^{2}} \cdot \frac{d}{d x} (5 x - 1)$

Step 3: Differentiate the inner function $(5 x - 1)$ with respect to $x$ : $\frac{d}{d x} (5 x - 1) = 5 - 0 = 5$

Step 4: Substitute back to obtain the final result:

$\frac{d y}{d x} = \frac{1}{1 - ( 5 x - 1 ) ^{2}} \cdot 5 = \frac{5}{1 - ( 5 x - 1 ) ^{2}}$

Thus, the derivative is:

$\frac{d y}{d x} = \frac{5}{1 - ( 5 x - 1 ) ^{2}}$

Key Formulas or Methods Used

Chain Rule: $\frac{d y}{d x} = \frac{d y}{d u} \cdot \frac{d u}{d x}$ for composite functions $y = f (g (x))$
Derivative of Inverse Sine: $\frac{d}{d x} [sin^{- 1} (x)] = \frac{1}{1 - x ^{2}}$
Power Rule: $\frac{d}{d x} [a x + b] = a$ for linear expressions

Summary of Steps

Identify the structure: Recognize $y = sin^{- 1} (5 x - 1)$ as a composite function with inner function $u = 5 x - 1$
Apply chain rule: Multiply the derivative of the outer function (evaluated at the inner function) by the derivative of the inner function
Differentiate outer function: Use $\frac{d}{d u} [sin^{- 1} (u)] = \frac{1}{1 - u ^{2}}$ with $u = 5 x - 1$
Differentiate inner function: Calculate $\frac{d}{d x} (5 x - 1) = 5$
Combine results: Multiply to get $\frac{5}{1 - ( 5 x - 1 ) ^{2}}$

Solution

We are given the function:

y = sin^{- 1} (5 x - 1)

To differentiate this with respect to

x

, we recognize this as a composite function where:

The outer function is

sin^{- 1} (u)

The inner function is

u = 5 x - 1

Step 1: Recall the derivative formula for inverse sine:

\frac{d}{d u} [sin^{- 1} (u)] = \frac{1}{1 - u ^{2}}

Step 2: Apply the chain rule, which states

\frac{d y}{d x} = \frac{d y}{d u} \cdot \frac{d u}{d x}

\frac{d y}{d x} = \frac{1}{1 - ( 5 x - 1 ) ^{2}} \cdot \frac{d}{d x} (5 x - 1)

Step 3: Differentiate the inner function

(5 x - 1)

with respect to

x

\frac{d}{d x} (5 x - 1) = 5 - 0 = 5

Step 4: Substitute back to obtain the final result:

\frac{d y}{d x} = \frac{1}{1 - ( 5 x - 1 ) ^{2}} \cdot 5 = \frac{5}{1 - ( 5 x - 1 ) ^{2}}

Thus, the derivative is:

\frac{d y}{d x} = \frac{5}{1 - ( 5 x - 1 ) ^{2}}

Summary of Steps

Identify the structure: Recognize

y = sin^{- 1} (5 x - 1)

as a composite function with inner function

u = 5 x - 1

Apply chain rule: Multiply the derivative of the outer function (evaluated at the inner function) by the derivative of the inner function

Differentiate outer function: Use

\frac{d}{d u} [sin^{- 1} (u)] = \frac{1}{1 - u ^{2}}

with

u = 5 x - 1

Differentiate inner function: Calculate

\frac{d}{d x} (5 x - 1) = 5

Combine results: Multiply to get

\frac{5}{1 - ( 5 x - 1 ) ^{2}}

2.6 Q-11

Question Statement

Background and Explanation

Solution

Key Formulas or Methods Used

Summary of Steps

2.6 Q-11

Question Statement

Background and Explanation

Solution

Key Formulas or Methods Used

Summary of Steps