Find the Derivative - d/dx xe^(-x^2)

Problem

(d(x)*e(−x2))/d(x)

Solution

1. Identify the rule needed for the derivative. Since the expression is a product of two functions, x and e(−x2) use the product rule: (d(u)*v)/d(x)=ud(v)/d(x)+vd(u)/d(x)

2. Assign the parts of the product rule where u=x and v=e(−x2)

3. Differentiate the first part, u=x

d(x)/d(x)=1

4. Differentiate the second part, v=e(−x2) using the chain rule. The derivative of eƒ(x) is eƒ(x)⋅ƒ(x)′

d(e(−x2))/d(x)=e(−x2)⋅(−2*x)

d(e(−x2))/d(x)=−2*x*e(−x2)

5. Apply the product rule formula by substituting the derivatives found in the previous steps.

(d(x)*e(−x2))/d(x)=x*(−2*x*e(−x2))+e(1)(−x2)

6. Simplify the expression by multiplying the terms and factoring out the common term e(−x2)

(d(x)*e(−x2))/d(x)=−2*x2*e(−x2)+e(−x2)

(d(x)*e(−x2))/d(x)=e(−x2)*(1−2*x2)

Final Answer

(d(x)*e(−x2))/d(x)=e(−x2)*(1−2*x2)

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