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

Problem

d()/d(x)*x7*ex

Solution

1. Identify the rule needed for the derivative. Since the expression is a product of two functions, x7 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=x7 and v=ex

3. Differentiate each part individually. The derivative of u is d(x7)/d(x)=7*x6 and the derivative of v is d(ex)/d(x)=ex

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

d(y)/d(x)=x7d(ex)/d(x)+exd(x7)/d(x)

5. Substitute the calculated derivatives into the expression.

d(y)/d(x)=x7*ex+ex*(7*x6)

6. Simplify the expression by factoring out the common terms x6 and ex

d(y)/d(x)=x6*ex*(x+7)

Final Answer

d()/d(x)*x7*ex=x6*ex*(x+7)

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