Find the Derivative - d/dx (1-4e^x)x^2

Problem

d()/d(x)*(1−4*ex)*x2

Solution

1. Identify the rule needed for the derivative of a product of two functions, u(x)=1−4*ex and v(x)=x2

2. Apply the product rule formula, which states d()/d(x)*[u⋅v]=u⋅d(v)/d(x)+v⋅d(u)/d(x)

3. Differentiate the individual components: d(x2)/d(x)=2*x and d(1−4*ex)/d(x)=−4*ex

4. Substitute these derivatives back into the product rule expression.

d()/d(x)*(1−4*ex)*x2=(1−4*ex)*(2*x)+(x2)*(−4*ex)

5. Distribute the terms to simplify the expression.

(1−4*ex)*(2*x)=2*x−8*x*ex

6. Combine all terms into a single expression.

2*x−8*x*ex−4*x2*ex

7. Factor out the common term 2*x if desired for the final form.

2*x*(1−4*ex−2*x*ex)

Final Answer

d()/d(x)*(1−4*ex)*x2=2*x−8*x*ex−4*x2*ex

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