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What is Derivative of e^(2x): Step-by-Step Guide

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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.

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).

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.

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