Question

If \(g\) is a differentiable function, find an expression for the

derivative of each of the following functions.

(a) \(y = xg\left( x \right)\) (b) \(y = \frac{x}{{g\left( x \right)}}\)

(c) \(y = \frac{{g\left( x \right)}}{x}\)

Short answer

(a) The required expression is \(y' = xg'\left( x \right) + g\left( x \right)\).

(b) The required expression is \(y' = \frac{{g\left( x \right) - xg'\left( x \right)}}{{{{\left( {g\left( x \right)} \right)}^2}}}\).

(c) The required expression is \(y' = \frac{{xg'\left( x \right) - g\left( x \right)}}{{{x^2}}}\).

Step-by-step solution

Derivative of the division of two function

If a function is a division of two functions we can differentiate the function by the following rule:

Let the function be \(h\left( x \right) = \frac{{f\left( x \right)}}{{g\left( x \right)}}\) then the derivativewill be,

\(\begin{aligned}h'\left( x \right) & = \frac{d}{{dx}}h\left( x \right)\\ & = \frac{d}{{dx}}\frac{{f\left( x \right)}}{{g\left( x \right)}}\\ & = \frac{{g\left( x \right)\frac{d}{{dx}} - f\left( x \right)\frac{d}{{dx}}g\left( x \right)}}{{{{\left( {g\left( x \right)} \right)}^2}}}\end{aligned}\)

Finding the derivative of the given function

(a) The function is given by \(y = xg\left( x \right)\).

Differentiate the function with respect to \(x\) we get,

\(\begin{aligned}y' & = \frac{d}{{dx}}\left( {xg\left( x \right)} \right)\\ & = xg'\left( x \right) + g\left( x \right)\end{aligned}\)

Hence \(y' = xg'\left( x \right) + g\left( x \right)\).

Finding the derivative of the given function

(b) The function is given by \(y = \frac{x}{{g\left( x \right)}}\) .

Differentiate the function with respect to \(x\) we get,

\(\begin{aligned}y' & = \frac{{g\left( x \right)\frac{{dx}}{{dx}} - x\frac{d}{{dx}}g\left( x \right)}}{{{{\left( {g\left( x \right)} \right)}^2}}}\\ & = \frac{{g\left( x \right) - xg'\left( x \right)}}{{{{\left( {g\left( x \right)} \right)}^2}}}\end{aligned}\)

Hence\(y' = \frac{{g\left( x \right) - xg'\left( x \right)}}{{{{\left( {g\left( x \right)} \right)}^2}}}\) .

Finding the derivative of the given function

(c) The function is given by \(y = \frac{{g\left( x \right)}}{x}\) .

Differentiate the function with respect to \(x\) we get,

\(\begin{aligned}y' & = \frac{{x\frac{{dg\left( x \right)}}{{dx}} - g\left( x \right)\frac{{dx}}{{dx}}}}{{{x^2}}}\\ & = \frac{{xg'\left( x \right) - g\left( x \right)}}{{{x^2}}}\end{aligned}\)

Hence \(y' = \frac{{xg'\left( x \right) - g\left( x \right)}}{{{x^2}}}\).