Question

11-18: Find the differential of the function.

11. \(y = {e^{5x}}\)

Short answer

The differential of the function is \(dy = 5{e^{5x}}dx\).

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 function is \(y = f\left( x \right) = {e^{5x}}\).

Evaluate the derivative of the function as shown below:

\(\begin{aligned}f'\left( x \right) &= \frac{d}{{dx}}\left( {{e^{5x}}} \right)\\ &= 5{e^{5x}}\end{aligned}\)

Obtain the differential of the function as shown below:

\(\begin{aligned}dy &= f'\left( x \right)dx\\dy &= 5{e^{5x}}dx\end{aligned}\)

Thus, the differential of the function is \(dy = 5{e^{5x}}dx\).