Questions: If f(x) = 4 sin^(-1)(x^2), find f'(x) Find f'(0.7).

Transcript text: If $f(x)=4 \sin ^{-1}\left(x^{2}\right)$, find $f^{\prime}(x)$ Find $f^{\prime}(0.7)$.

Solution

Solution Steps

To find the derivative of the function $ f(x) = 4 \sin^{-1}(x^2) $, we will use the chain rule. The chain rule states that if you have a composite function, the derivative is the derivative of the outer function evaluated at the inner function times the derivative of the inner function.

Identify the outer function and the inner function.
Differentiate the outer function with respect to the inner function.
Differentiate the inner function with respect to $ x $.
Multiply the results from steps 2 and 3 to get the derivative $ f'(x) $.

Next, we will evaluate $ f'(x) $ at $ x = 0.7 $.

Step 1: Define the Function

Given the function $ f(x) = 4 \sin^{-1}(x^2) $.

Step 2: Differentiate the Function

To find the derivative $ f'(x) $, we use the chain rule. The outer function is $ 4 \sin^{-1}(u) $ where $ u = x^2 $. The derivative of $ \sin^{-1}(u) $ is $ \frac{1}{\sqrt{1 - u^2}} $, and the derivative of $ u = x^2 $ is $ 2x $.

Thus, the derivative $ f'(x) $ is: \[ f'(x) = 4 \cdot \frac{1}{\sqrt{1 - (x^2)^2}} \cdot 2x = \frac{8x}{\sqrt{1 - x^4}} \]

Step 3: Evaluate the Derivative at $ x = 0.7 $

Substitute $ x = 0.7 $ into the derivative: \[ f'(0.7) = \frac{8 \cdot 0.7}{\sqrt{1 - (0.7)^4}} = \frac{5.6}{\sqrt{1 - 0.2401}} = \frac{5.6}{\sqrt{0.7599}} \]

Using a calculator, we find: \[ f'(0.7) \approx 6.424 \]