Find the Derivative - d/dx y=x^7e^x

Problem

d()/d(x)*x^7*e^x

Solution

1. Identify the rule needed for the derivative. Since the expression is a product of two functions, x^7 and e^x 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=x^7 and v=e^x

3. Differentiate each part individually. The derivative of u is d(x^7)/d(x)=7*x^6 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^7d(e^x)/d(x)+e^xd(x^7)/d(x)

5. Substitute the calculated derivatives into the expression.

d(y)/d(x)=x^7*e^x+e^x*(7*x^6)

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

d(y)/d(x)=x^6*e^x*(x+7)

Final Answer

d()/d(x)*x^7*e^x=x^6*e^x*(x+7)

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