Find the Derivative - d/dx y=x^6e^(-4x)

Problem

d()/d(x)*x6*e(−4*x)

Solution

1. Identify the rule needed for the derivative of a product of two functions, u=x6 and v=e(−4*x)

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

3. Differentiate the first part, u=x6 using the power rule to get d(x6)/d(x)=6*x5

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

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

d(y)/d(x)=x6*(−4*e(−4*x))+e(−4*x)*(6*x5)

6. Simplify the expression by factoring out common terms, specifically 2*x5*e(−4*x)

d(y)/d(x)=−4*x6*e(−4*x)+6*x5*e(−4*x)

d(y)/d(x)=2*x5*e(−4*x)*(3−2*x)

Final Answer

(d(x6)*e(−4*x))/d(x)=2*x5*e(−4*x)*(3−2*x)

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