Find dy/dx y=x^6e^x

Problem

d()/d(x)*x6*ex

Solution

1. Identify the rule needed for the derivative. Since the expression is a product of two functions, x6 and ex use the product rule.

2. Apply the product rule formula, which states that (d(u)*v)/d(x)=ud(v)/d(x)+vd(u)/d(x)

3. Assign the variables such that u=x6 and v=ex

4. Differentiate each part individually.

d(x6)/d(x)=6*x5

d(ex)/d(x)=ex

5. Substitute these derivatives back into the product rule formula.

d(y)/d(x)=x6*ex+ex*(6*x5)

6. Factor out the common terms x5 and ex to simplify the expression.

d(y)/d(x)=x5*ex*(x+6)

Final Answer

d(y)/d(x)=x6*ex+6*x5*ex

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