Find the Derivative - d/dx y=x^10e^x

Problem

d()/d(x)*x^10*e^x

Solution

1. Identify the rule needed for the derivative. Since the expression is a product of two functions, x^10 and e^x use the product rule: (d(u)*v)/d(x)=ud(v)/d(x)+vd(u)/d(x)

2. Assign the functions for the product rule. Let u=x^10 and v=e^x

3. Differentiate each part individually. The derivative of x^10 is 10*x^9 using the power rule, and the derivative of e^x is e^x

4. Apply the product rule formula by substituting the functions and their derivatives.

(d(x^10)*e^x)/d(x)=x^10d(e^x)/d(x)+e^xd(x^10)/d(x)

5. Substitute the derivatives into the expression.

(d(x^10)*e^x)/d(x)=x^10*e^x+e^x*(10*x^9)

6. Factor out the common terms x^9 and e^x to simplify the final expression.

(d(x^10)*e^x)/d(x)=x^9*e^x*(x+10)

Final Answer

(d(x^10)*e^x)/d(x)=x^9*e^x*(x+10)

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