Question

similar question | all parts showi differentiate.
( y = \frac{2}{1 - 7x^2} )
( \frac{dy}{dx} = -\frac{56x}{(1 - 7x^2)^{-3}} )

Explanation:

Step1: Rewrite the function

Rewrite \( y = \frac{2}{1 - 7x^2} \) as \( y = 2(1 - 7x^2)^{-1} \).

Step2: Apply the chain rule

The chain rule states that if \( y = f(g(x)) \), then \( y' = f'(g(x)) \cdot g'(x) \). Let \( u = 1 - 7x^2 \), so \( y = 2u^{-1} \). First, differentiate \( y \) with respect to \( u \): \( \frac{dy}{du} = 2(-1)u^{-2} = -2u^{-2} \). Then, differentiate \( u \) with respect to \( x \): \( \frac{du}{dx} = -14x \).

Step3: Multiply the derivatives

By the chain rule, \( \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} \). Substitute the values: \( \frac{dy}{dx} = -2u^{-2} \cdot (-14x) \). Replace \( u \) with \( 1 - 7x^2 \): \( \frac{dy}{dx} = -2(1 - 7x^2)^{-2} \cdot (-14x) = 28x(1 - 7x^2)^{-2} = \frac{28x}{(1 - 7x^2)^2} \). (Note: The given answer in the image seems incorrect. The correct derivative using the chain rule is as above.)

Answer:

\( \frac{28x}{(1 - 7x^2)^2} \) (or equivalent form \( 28x(1 - 7x^2)^{-2} \))