\(Differentiate f (x) = \frac{1}{(1 - x^2)^5}\) with respect to \(x\).
FURTHER MATHEMATICS
WAEC 2023
\(Differentiate f (x) = \frac{1}{(1 - x^2)^5}\) with respect to \(x\).
- A. \(\frac{-5x}{(1-x^2)^6}\)
- B. \(\frac{-10x}{(1-x^2)^6}\)
- C. \(\frac{5x}{(1-x^2)^6}\)
- D. \(\frac{10x}{(1-x^2)^6}\)
Correct Answer: D. \(\frac{10x}{(1-x^2)^6}\)
Explanation
\(y = \frac{1} {(1 - x^2)^5} = (1-x^2)^{-5}\)
Let u = \(1 - x^2; y = u ^{-5}\)
\(\frac{du}{ dx}=-2x;\frac{dy}{ du}=-5u^{-6}\)
Using chain rule:
\(\frac{dy}{dx}=\frac{dy}{du}\times\frac{du}{ dx}\)
\(\frac{dy}{dx}=-2x\times-5u^{-6}\)
\(\frac{dy}{dx}=-2x\times-5(1 - x^2)^{-6}\)
\(\therefore\frac{dy}{dx} = 10x(1-x^2)^{-6}=\frac{10x}{(1-x^2)^6}\)
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