Find dy/dx y=x^8e^x

Problem

d()/d(x)*x^8*e^x

Solution

1. Identify the rule needed for the derivative. Since the expression is a product of two functions, x^8 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^8 and v=e^x

3. Differentiate each part individually. The derivative of u is d(x^8)/d(x)=8*x^7 and the derivative of v is d(e^x)/d(x)=e^x

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

d(y)/d(x)=x^8d(e^x)/d(x)+e^xd(x^8)/d(x)

5. Substitute the calculated derivatives into the expression.

d(y)/d(x)=x^8*e^x+e^x*(8*x^7)

6. Simplify the expression by factoring out the common terms x^7 and e^x

d(y)/d(x)=x^7*e^x*(x+8)

Final Answer

d(y)/d(x)=x^8*e^x+8*x^7*e^x

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