Learn how to find the derivative of e^(xy). The derivative is d/dx[e^(xy)].
Below is the graph of f(x) = e^(xy) and its derivative f'(x). Notice how the original function and derivative relate to each other.
Original Function:
Derivative:
This derivative represents the rate of change of the original function. The calculation follows standard differentiation rules as shown in the step-by-step solution below.
The function e^(xy) involves an exponential with a product of two variables, x and y, in the exponent. To differentiate this, we need to apply the Power Rule for derivatives, which is appropriate when dealing with exponential functions raised to a power.
💡 Why this works: By recognizing the structure e^(xy), we understand that the differentiation must consider how both variables, x and y, interact within the exponent. The product rule is indirectly used here due to the involvement of both variables in the exponent.
To differentiate e^(xy), apply the Power Rule for the exponential function while using the chain rule to handle the product of x and y in the exponent.
💡 Why this works: The derivative of e^(xy) is found by first differentiating the exponential function and then differentiating the exponent, xy, with respect to x. The product rule applies when differentiating the exponent, resulting in a derivative that accounts for the change in x and y.
After applying the differentiation rules, simplify the expression to obtain the final form of the derivative. Ensure that the application of the Power Rule is correct and consistent.
💡 Why this works: The final derivative, d/dx[e^(xy)], is the correct form, confirming the proper application of the Power Rule and chain rule. The simplification step ensures the expression is fully accurate and ready for use.
[Teaching explanation - to be filled]
[Application to this function - to be filled]
❌ Common Mistake:
Not simplifying the final answer
✅ Correct Approach:
Always simplify your derivative to its most reduced form
Double-check your work by comparing your steps with our solution. Use our calculator to verify each step of the differentiation process.
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