Find the Derivative - d/dx e^(3x)cos(2x)

Problem

d()/d(x)*e(3*x)*cos(2*x)

Solution

1. Identify the rule needed for the derivative of a product of two functions, u(x)=e(3*x) and v(x)=cos(2*x)

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

3. Differentiate the first part using the chain rule: d(e(3*x))/d(x)=3*e(3*x)

4. Differentiate the second part using the chain rule: d(cos(2*x))/d(x)=−2*sin(2*x)

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

(d(e(3*x))*cos(2*x))/d(x)=e(3*x)*(−2*sin(2*x))+cos(2*x)*(3*e(3*x))

6. Simplify the expression by factoring out the common term e(3*x)

(d(e(3*x))*cos(2*x))/d(x)=e(3*x)*(3*cos(2*x)−2*sin(2*x))

Final Answer

(d(e(3*x))*cos(2*x))/d(x)=e(3*x)*(3*cos(2*x)−2*sin(2*x))

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