Question

3-34 Differentiate the function.

11. \(f\left( x \right) = {x^{\frac{3}{2}}} + {x^{ - 3}}\)

Short answer

The differentiation of the function \(f\left( x \right) = {x^{\frac{3}{2}}} + {x^{ - 3}}\) is \(f'\left( x \right) = \frac{3}{2}{x^{\frac{1}{2}}} - 3{x^{ - 4}}\).

Step-by-step solution

Write the formula of the sum rules and the power rule

The Sum Rule:\(\frac{d}{{dx}}\left( {f\left( x \right) + g\left( x \right)} \right) = \frac{d}{{dx}}f\left( x \right) + \frac{d}{{dx}}g\left( x \right)\)

The Power Rule: \(\frac{d}{{dx}}\left( {{x^n}} \right) = n{x^{n - 1}}\) where \(n\) is a positive integer

Find the differentiation of the function

Consider the function \(f\left( x \right) = {x^{\frac{3}{2}}} + {x^{ - 3}}\). Differentiate the function w.r.t \(x\) by using the sum rule and the power rule.

\(\begin{aligned}\frac{{d\left( {f\left( x \right)} \right)}}{{dx}} &= \frac{{d\left( {{x^{\frac{3}{2}}} + {x^{ - 3}}} \right)}}{{dx}}\\ &= \frac{d}{{dx}}\left( {{x^{\frac{3}{2}}}} \right) + \frac{d}{{dx}}\left( {{x^{ - 3}}} \right)\\ &= \frac{d}{{dx}}\left( {\frac{3}{2}{x^{\frac{3}{2} - 1}}} \right) + \frac{d}{{dx}}\left( { - 3{x^{ - 3 - 1}}} \right)\\ &= \frac{3}{2}{x^{\frac{1}{2}}} - 3{x^{ - 4}}\end{aligned}\)

Thus, the derivative of the function \(f\left( x \right) = {x^{\frac{3}{2}}} + {x^{ - 3}}\) is \(f'\left( x \right) = \frac{3}{2}{x^{\frac{1}{2}}} - 3{x^{ - 4}}\).