Differentiation using the product rule

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

Express the denominator as a negative power

\[ \begin{align} &\frac{d}{dx}\frac{(2x+1)^2}{x} \\ &= \frac{d}{dx}((2x+1)^2 \times x^{-1}) \\ &= \frac{d}{dx}(2x+1)^2 \times x^{-1} + (2x+1)^2 \times \frac{d}{dx} x^{-1}\\ &= 4(2x+1) x^{-1} - (2x+1)^2 \times x^{-2} \end{align} \ \] Note the use of the chain rule to differentiate the \( (2x+1)^2 \) term.

Simplify the result.

\[ \begin{align} &= \frac{(2x+1)(4x - (2x+1))}{x^2} \\ &= \frac{(2x + 1)(2x - 1)} {x^2} \\ \end{align} \ \]

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
\[ \frac{d}{dx}\frac{(5x+3)^3}{x^2} \]