Differentiating products

 

Evaluate
\[ \frac{d}{dx} (\sqrt{x^2+1} \times \sqrt{(x+1)^3}\ ) \quad \ \]

Re-express the square roots as numeric powers before differentiating.

\[ \begin{align} \frac{d}{dx} (x^2+1)^{1/2} \times (x+1)^{3/2} \end{align} \ \]

Using the product rule gives

\[ \begin{align} &\frac{d}{dx} (x^2+1)^{1/2} \times (x+1)^{3/2} \\ &= -\frac{1}{2}(x^2+1)^{-1/2} 2x \times (x+1)^{3/2} \\ &+ (x^2+1)^{1/2} \times \frac{3}{2}(x+1)^{1/2} \times 1 \\ \end{align} \ \]

Tidy the expression

\[ \begin{align} &= -x(x^2+1)^{-1/2} \times (x+1)^{3/2} \\ &+ (x^2+1)^{1/2} \times \frac{3}{2}(x+1)^{1/2} \\ &= \frac{3}{2} \sqrt{x^2+1} \times \sqrt{x+1} - \frac{x \sqrt{(x+1)^3}}{\sqrt{x^2+1}} \end{align} \ \]

You will gain most benefit if you work through each problem first and then come back and check your work against the solution.

 

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Now You Try:

Evaluate
\[ \frac{d}{dx} (\sqrt{2x+3} \times \sqrt{(x+1)^2}\ ) \quad \ \]