Question

Find the differential of the function.

15. \(y = \frac{1}{{{x^2} - 3x}}\)

Short answer

The differential of the function is \(dy = - \frac{{2x - 3}}{{{{\left( {{x^2} - 3x} \right)}^2}}}dx\).

Step-by-step solution

Definition of differentials

The equation establishes the differential \(dy\)with respect to \(dx\) as shown below:

\(dy = f'\left( x \right)dx\)

Where \(dx\) represent the independent variableand \(dy\) represent the dependent variable.

Determine the differential of the function

The given function is \(y = f\left( x \right) = \frac{1}{{{x^2} - 3x}}\).

Rewrite the function as shown below:

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

Evaluate the derivative of the function as shown below:

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

Obtain the differential of the function as shown below:

\(\begin{aligned}dy &= f'\left( x \right)dx\\dy &= - \frac{{2x - 3}}{{{{\left( {{x^2} - 3x} \right)}^2}}}dx\end{aligned}\)

Thus, the differential of the function is \(dy = - \frac{{2x - 3}}{{{{\left( {{x^2} - 3x} \right)}^2}}}dx\).