Question

Find the derivative of the function

\(f\left( x \right) = \frac{{{x^{\bf{4}}} - {\bf{5}}{x^{\bf{3}}} + \sqrt x }}{{{x^{\bf{2}}}}}\)

in two ways: by using the Quotient Rule and by simplifying first.Show that your answers are equivalent. Which method do you prefer?

Short answer

By quotient rule: \(f'\left( x \right) = 2x - 5 - \frac{3}{2}{x^{ - \frac{5}{2}}}\)

By simplifying first: \(f'\left( x \right) = 2x - 5 - \frac{3}{2}{x^{ - \frac{5}{2}}}\)

The expression of derivative is same with both the method. The method of simplifying first is more easier.

Step-by-step solution

Step 1:Find the derivative of f using the product rule

The equation for Quotient rule is \({\left( {\frac{f}{g}} \right)'} = \frac{{fg' - gf'}}{{{g^2}}}\).

Apply product rule for the function \(f\left( x \right) = \frac{{{x^4} - 5{x^3} + \sqrt x }}{{{x^2}}}\).

\(\begin{aligned}f'\left( x \right) &= \frac{{\rm{d}}}{{{\rm{d}}x}}\left( {\frac{{{x^4} - 5{x^3} + \sqrt x }}{{{x^2}}}} \right)\\ &= \frac{{{x^2}\frac{{\rm{d}}}{{{\rm{d}}x}}\left( {{x^4} - 5{x^3} + \sqrt x } \right) - \left( {{x^4} - 5{x^3} + \sqrt x } \right)\frac{{\rm{d}}}{{{\rm{d}}x}}\left( {{x^2}} \right)}}{{{{\left( {{x^2}} \right)}^2}}}\\ &= \frac{{{x^2}\left( {4{x^3} - 15{x^2} + \frac{1}{{2\sqrt x }}} \right) - 2x\left( {{x^4} - 5{x^3} + \sqrt x } \right)}}{{{x^4}}}\\ &= \frac{{4{x^4} - 15{x^3} + \frac{1}{2}\sqrt x - 2{x^4} + 10{x^3} + 2\sqrt x }}{{{x^3}}}\\ &= \frac{{2{x^4} - 5{x^3} + \frac{3}{2}\sqrt x }}{{{x^3}}}\\ &= 2x - 5 - \frac{3}{2}{x^{ - \frac{5}{2}}}\end{aligned}\)

Therefore, \(f'\left( x \right) = 2x - 5 - \frac{3}{2}{x^{ - \frac{5}{2}}}\) by quotient rule.

Simplify the function \(f\left( x \right) = \frac{{{x^{\bf{4}}} - {\bf{5}}{x^{\bf{3}}} + \sqrt x }}{{{x^{\bf{2}}}}}\)

The function\(f\left( x \right) = \frac{{{x^4} - 5{x^3} + \sqrt x }}{{{x^2}}}\) can be simplified as,

\(\begin{aligned}f\left( x \right) &= \frac{{{x^4} - 5{x^3} + \sqrt x }}{{{x^2}}}\\ &= {x^2} - 5x + {x^{ - \frac{3}{2}}}\end{aligned}\)

Differentiate the function \(f\left( x \right) = {x^{\bf{2}}} - {\bf{5}}x + {x^{ - \frac{{\bf{3}}}{{\bf{2}}}}}\)

Differentaite the function \(f\left( x \right) = {x^2} - 5x + {x^{ - \frac{3}{2}}}\).

\(\begin{aligned}f'\left( x \right) &= \frac{{\rm{d}}}{{{\rm{d}}x}}\left( {{x^2} - 5x + {x^{ - \frac{3}{2}}}} \right)\\ &= 2x - 5 - \frac{3}{2}{x^{ - \frac{5}{2}}}\end{aligned}\)

Therefore, \(f'\left( x \right) = 2x - 5 - \frac{3}{2}{x^{ - \frac{5}{2}}}\) by simplifying first.

Thus, both the methods agrees to derivative for same answer. The method of simplifying first seems to be more easier.