Find the Derivative - d/dx y=x^9e^x

Problem

d()/d(x)*x9*ex

Solution

1. Identify the rule needed for the derivative. Since the expression is a product of two functions, x9 and ex use the product rule: (d(u)*v)/d(x)=ud(v)/d(x)+vd(u)/d(x)

2. Assign the functions to the variables u and v Let u=x9 and v=ex

3. Differentiate each part individually. The derivative of u is d(x9)/d(x)=9*x8 using the power rule. The derivative of v is d(ex)/d(x)=ex

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

(d(x9)*ex)/d(x)=x9d(ex)/d(x)+exd(x9)/d(x)

5. Substitute the calculated derivatives into the expression.

(d(x9)*ex)/d(x)=x9*ex+ex*(9*x8)

6. Simplify the expression by factoring out the greatest common factor, which is x8*ex

(d(x9)*ex)/d(x)=x8*ex*(x+9)

Final Answer

(d(x9)*ex)/d(x)=x8*ex*(x+9)

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