Find the Derivative - d/dx y=6x^3e^(-x)

Problem

d()/d(x)*6*x3*e(−x)

Solution

1. Identify the rule needed for the expression, which is the product rule for two functions u=6*x3 and v=e(−x)

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

3. Differentiate the first part u=6*x3 using the power rule to get d(u)/d(x)=18*x2

4. Differentiate the second part v=e(−x) using the chain rule to get d(v)/d(x)=−e(−x)

5. Substitute these derivatives back into the product rule formula.

(d(6)*x3*e(−x))/d(x)=(6*x3)*(−e(−x))+(e(−x))*(18*x2)

6. Simplify the expression by factoring out common terms, specifically 6*x2*e(−x)

(d(6)*x3*e(−x))/d(x)=−6*x3*e(−x)+18*x2*e(−x)

(d(6)*x3*e(−x))/d(x)=6*x2*e(−x)*(3−x)

Final Answer

(d(6)*x3*e(−x))/d(x)=6*x2*e(−x)*(3−x)

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