Differentiation using the product rule

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

Use the term in x squared as the first part of the product. Then

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

Simplify the expression.

\[ \begin{align} &= \frac{4x(x+k) + x^2}{2\ \sqrt{x+k}} \\ &= \frac{5x^2 + 4kx}{2\ \sqrt{x+k}} \\ \end{align} \ \]

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