Differentiation using the product rule

Evaluate
\[ \frac{d}{dx} x \sqrt{(1-x^2)} \]

Use the square root term as the second part of the product

\[ \begin{align} &\frac{d}{dx} (x \sqrt{(1-x^2)}) \\ &= \frac{d}{dx}(x)\times \sqrt{1-x^2} + x \frac{d}{dx} (\sqrt{1-x^2}) \\ &= \sqrt{1-x^2} + \frac{x}{2\sqrt{1-x^2}}\frac{d}{dx}(1-x^2) \\ &= \sqrt{1-x^2} + \frac{x(-2x)}{2\sqrt{1-x^2}} \\ &= \sqrt{1-x^2} + \frac{-x^2}{\sqrt{1-x^2}} \end{align} \ \] Note the use of the chain rule for differentiating the square root.

Simplify the expression by putting everything over a common denominator.

\[ \begin{align} &= \frac{1 - x^2 - x^2}{\sqrt{1-x^2}} \\ &= \frac{1 - 2x^2}{\sqrt{1-x^2}} \\ \end{align} \ \]

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What term would you use as the second part of the product?

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Now You Try:

Evaluate
\[ \frac{d}{dx} x^2 \sqrt{(x^2+1)} \]