Question

Find the derivative of the function:

24. \(y = {\left( {x + \frac{1}{x}} \right)^5}\)

Short answer

The derivative of \(y\) is \(5{\left( {x + \frac{1}{x}} \right)^4}\left( {1 - \frac{1}{{{x^2}}}} \right)\).

Step-by-step solution

The chain rule

If \(g\) is differentiable at \(x\) and \(f\) is differentiable at \(g\left( x \right)\), then the composite function \(F = f \circ g\) defined by \(F\left( x \right) = f\left( {g\left( x \right)} \right)\) is differentiable at \(x\) and \(F'\) is given by the product \(F'\left( x \right) = f'\left( {g\left( x \right)} \right) \cdot g'\left( x \right)\). In Leibniz notation, if \(y = f\left( u \right)\) and \(u = g\left( x \right)\) are both differentiable functions, then \(\frac{{dy}}{{dx}} = \frac{{dy}}{{du}} \cdot \frac{{du}}{{dx}}\).

The derivative of the given function

Differentiate \(y\) with respect to \(x\) as follows:

\(\begin{aligned}\frac{{dy}}{{dx}}&= \frac{d}{{dx}}\left( {{{\left( {x + \frac{1}{x}} \right)}^5}} \right)\\ &= 5{\left( {x + \frac{1}{x}} \right)^4}\frac{d}{{dx}}\left( {x + \frac{1}{x}} \right)\\ &= 5{\left( {x + \frac{1}{x}} \right)^4}\left( {1 - \frac{1}{{{x^2}}}} \right)\end{aligned}\)

Hence, the derivative of \(y\) is \(5{\left( {x + \frac{1}{x}} \right)^4}\left( {1 - \frac{1}{{{x^2}}}} \right)\).