Find the Derivative - d/dx y=x^12e^x

Problem

d()/d(x)*x^12*e^x

Solution

1. Identify the rule needed for the derivative. Since the expression is a product of two functions, u=x^12 and v=e^x use the product rule.

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

3. Differentiate the individual components. The derivative of x^12 is 12*x^11 using the power rule, and the derivative of e^x is e^x

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

(d(x^12)*e^x)/d(x)=x^12d(e^x)/d(x)+e^xd(x^12)/d(x)

5. Simplify the resulting expression by performing the differentiation and factoring out common terms.

(d(x^12)*e^x)/d(x)=x^12*e^x+e^x*(12*x^11)

(d(x^12)*e^x)/d(x)=x^11*e^x*(x+12)

Final Answer

(d(x^12)*e^x)/d(x)=x^11*e^x*(x+12)

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