Find the Derivative - d/dx 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 the product rule must be applied: (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=x6 and v=ex

3. Differentiate each part individually. The derivative of x6 is 6*x5 using the power rule, and the derivative of ex is ex

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

(d(x6)*ex)/d(x)=x6d(ex)/d(x)+exd(x6)/d(x)

5. Substitute the calculated derivatives into the expression.

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

6. Simplify the expression by factoring out the greatest common factor, which is x5*ex

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

Final Answer

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

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