Find the differential of the function
[tex]
\begin{array}{c}
y=e^{4 x} \\
d y=\square
\end{array}
[/tex]
Answer (1)
Find the derivative of y = e 4 x using the chain rule: d x d y = 4 e 4 x .
Express the differential as d y = d x d y d x .
Substitute the derivative into the differential equation: d y = 4 e 4 x d x .
The differential of the function is 4 e 4 x d x .
Explanation
Problem Analysis We are given the function y = e 4 x and we want to find its differential d y . The differential d y is related to the derivative d x d y by the formula d y = d x d y d x . So, we first need to find the derivative of y with respect to x .
Finding the Derivative To find the derivative of y = e 4 x with respect to x , we use the chain rule. The chain rule states that if y = f ( g ( x )) , then d x d y = f ′ ( g ( x )) g ′ ( x ) . In our case, f ( u ) = e u and g ( x ) = 4 x . Thus, f ′ ( u ) = e u and g ′ ( x ) = 4 . Applying the chain rule, we have d x d y = e 4 x ⋅ 4 = 4 e 4 x .
Finding the Differential Now that we have the derivative, we can find the differential d y . We know that d y = d x d y d x . Substituting the derivative we found in the previous step, we get d y = 4 e 4 x d x .
Final Answer Therefore, the differential of the function y = e 4 x is d y = 4 e 4 x d x .
Examples
In physics, if x represents time and y = e 4 x represents the population of a bacteria colony, then d y approximates the change in population over a small change in time d x . For instance, if at a certain time x , the population is e 4 x , then for a small time increment d x , the change in population is approximately 4 e 4 x d x . This is useful for modeling exponential growth in various real-world scenarios.
