N-th Derivative of x e^x
Logarithmic and exponential functions
core
Prove that \( D_{x}^{n}(xe^{x}) = (x+n)e^{x} \).
calculus-idifferentiationhigher-order-derivativesnth-derivativemathematical-inductionproof-by-inductionproduct-ruleexponential-functiontranscendental-functions
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What you need first
- Mathematical induction
- The product rule for differentiation
- The derivative of the exponential function
What this problem tests
Tests the ability to apply the product rule repeatedly and to structure a formal proof by mathematical induction.
This is a classic problem in introductory calculus courses, often used to introduce the concept of finding patterns in higher-order derivatives and proving them with induction.
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