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Find the derivative of f(x) f(x) .\newlinef(x)=xex f(x) = x e^x \newlinef′(x)= f'\left(x\right) = ______

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Q. Find the derivative of f(x) f(x) .\newlinef(x)=xex f(x) = x e^x \newlinef′(x)= f'\left(x\right) = ______
  1. Identify Functions: First, identify the two functions that are being multiplied in f(x)=xexf(x) = xe^x. The first function is xx and the second function is exe^x.
  2. Find Derivative of xx: Next, find the derivative of the first function, which is xx. The derivative of xx with respect to xx is 11.
  3. Find Derivative of exe^x: Then, find the derivative of the second function, which is exe^x. The derivative of exe^x with respect to xx is exe^x itself.
  4. Apply Product Rule: Now, apply the product rule for differentiation. The product rule states that the derivative of a product of two functions is the derivative of the first function times the second function plus the first function times the derivative of the second function. So, f′(x)=(1)(ex)+(x)(ex)f'(x) = (1)(e^x) + (x)(e^x).
  5. Simplify Expression: Simplify the expression obtained in the previous step. f′(x)=ex+xexf'(x) = e^x + xe^x.

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