Find the Derivative - d/dx y=3^xx^3

Problem

d()/d(x)*3^x*x^3

Solution

1. Identify the rule needed for the derivative. Since the expression is a product of two functions, 3^x and x^3 apply the product rule: (d(u)*v)/d(x)=ud(v)/d(x)+vd(u)/d(x)

2. Differentiate the first part of the product. The derivative of the exponential function 3^x is 3^x*ln(3)

3. Differentiate the second part of the product. Using the power rule, the derivative of x^3 is 3*x^2

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

(d(3^x)*x^3)/d(x)=3^xd(x^3)/d(x)+x^3d(3^x)/d(x)

5. Substitute the calculated derivatives into the expression.

(d(3^x)*x^3)/d(x)=3^x*(3*x^2)+x^3*(3^x*ln(3))

6. Factor out the common terms 3^x and x^2 to simplify the final expression.

(d(3^x)*x^3)/d(x)=3^x*x^2*(3+x*ln(3))

Final Answer

(d(3^x)*x^3)/d(x)=3^x*x^2*(3+x*ln(3))

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