In the exploration of calculus, finding derivatives forms a fundamental aspect, unveiling the rate of change within functions. Here, we embark on a journey to unravel the derivative of e^2x employing the chain rule, a cornerstone principle in calculus. Let’s solve this mathematical puzzle, layer by layer, to find its solution.
Given the function y = e^2x, we want to find its derivative dy/dx.
Step 1: Begin by recognizing that the function y can be written as
y = e^u, where u = 2x.
Step 2: Applying the chain rule, we have
dy/dx = du/dx × dy/du.
Step 3: Now we need to find du/dx,
The derivative of u with respect to x, which equals 2.
Step 4: Since y = eu, differentiate y with respect to u,
dy/du = eu.
Step 5: Substituting the derivatives and values into the chain rule expression, we get
dy/dx = dy/du × du/dx.
Step 6: Simplifying, we have
dy/dx = 2(e^2x).
Thus,
Conclusion
Through meticulous application of the chain rule, we’ve unveiled the derivative of e^2x to be 2(e^2x).
This solution, albeit expressed with differing notation, harmonizes with the previous explanation, reaffirming the robustness of calculus principles. By embracing alternate pathways, we deepen our understanding of calculus concepts, forging a stronger foundation for future explorations.
We hope this article has been helpful. If you have any questions, please feel free to comment below.
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