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

Problem

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

Solution

1. Identify the constant multiple rule and the product rule. The constant 3 can be moved outside the derivative, and the function is a product of u=e^x and v=cos(x)

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

3. Differentiate the individual components. The derivative d(e^x)/d(x)=e^x and the derivative d(cos(x))/d(x)=−sin(x)

4. Substitute these derivatives back into the product rule expression.

3*(e^x*(−sin(x))+cos(x)*e^x)

5. Simplify the expression by factoring out the common term e^x

3*e^x*(cos(x)−sin(x))

Final Answer

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

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