Find the Derivative - d/dx y=x^2e^x-2xe^x+2e^x

Problem

d()/d(x)*(x2*ex−2*x*ex+2*ex)

Solution

1. Apply the sum rule for differentiation by taking the derivative of each term individually.

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

2. Apply the product rule to the first term x2*ex where (d(u)*v)/d(x)=ud(v)/d(x)+vd(u)/d(x)

(d(x2)*ex)/d(x)=x2*ex+2*x*ex

3. Apply the product rule to the second term 2*x*ex

(d(2)*x*ex)/d(x)=2*x*ex+2*ex

4. Differentiate the third term 2*ex using the rule for exponential functions.

(d(2)*ex)/d(x)=2*ex

5. Combine the results and simplify the expression by grouping like terms.

d(y)/d(x)=(x2*ex+2*x*ex)−(2*x*ex+2*ex)+2*ex

6. Simplify the equation by canceling out terms that sum to zero.

d(y)/d(x)=x2*ex+2*x*ex−2*x*ex−2*ex+2*ex

d(y)/d(x)=x2*ex

Final Answer

d()/d(x)*(x2*ex−2*x*ex+2*ex)=x2*ex

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