Find the Derivative - d/dx x^6e^x

Problem

d()/d(x)*x^6*e^x

Solution

1. Identify the rule needed for the derivative. Since the expression is a product of two functions, x^6 and e^x 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=x^6 and v=e^x

3. Differentiate each part individually. The derivative of x^6 is 6*x^5 using the power rule, and the derivative of e^x is e^x

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

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

5. Substitute the calculated derivatives into the expression.

(d(x^6)*e^x)/d(x)=x^6*e^x+e^x*(6*x^5)

6. Simplify the expression by factoring out the greatest common factor, which is x^5*e^x

(d(x^6)*e^x)/d(x)=x^5*e^x*(x+6)

Final Answer

(d(x^6)*e^x)/d(x)=x^5*e^x*(x+6)

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