Question

11-18: Find the differential of the function

18. \(y = \frac{{{e^x}}}{{1 - {e^x}}}\)

Short answer

The differential of the function is \(dy = \frac{{{e^x}}}{{{{\left( {1 - {e^x}} \right)}^2}}}dx\).

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 theindependent variableand \(dy\) represent thedependent variable.

Determine the differential of the function

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

Use quotient rule to evaluate the derivative of the function as shown below:

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

Obtain the differential of the function as shown below:

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

Thus, the differential of the function is \(dy = \frac{{{e^x}}}{{{{\left( {1 - {e^x}} \right)}^2}}}dx\).