Stop falling for the variable exponent trap! 💥
When finding the derivative of x^x, both the Power Rule and Exponential Rule fail because both the base and exponent are variables!
Here is how to solve it in seconds using Logarithmic Differentiation:
Problem: Find d/dx (x^x)
1️⃣ Why Standard Rules Fail:
• Power Rule gives x · x^(x-1) 🛑 (WRONG)
• Exponential Rule gives x^x · ln(x) 🛑 (WRONG)
2️⃣ Logarithmic Differentiation:
• Set y = x^x
• Take natural log of both sides: ln(y) = x · ln(x)
• Differentiate implicitly: (1/y)(dy/dx) = 1 + ln(x)
• Multiply by y: dy/dx = x^x (1 + ln(x)) ✨
Memorize this formula: d/dx (x^x) = x^x (1 + ln(x))!
🧠 YOUR TURN:
What is the value of d/dx (x^x) at x = 1?
(Hint: Plug x = 1 into x^x (1 + ln(x)) and remember that ln(1) = 0!)
Drop your answer in the comments below! 👇
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