Find the Derivative - d/dx 6e^xcos(x)

Problem

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

Solution

1. Identify the constant multiple rule to move the constant 6 outside the derivative.

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

2. Apply the product rule, which states that (d(u)*v)/d(x)=ud(v)/d(x)+vd(u)/d(x) where u=e^x and v=cos(x)

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

3. Differentiate the individual functions using the rules d(cos(x))/d(x)=−sin(x) and d(e^x)/d(x)=e^x

6*(e^x*(−sin(x))+cos(x)*e^x)

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

6*e^x*(cos(x)−sin(x))

Final Answer

(d(6)*e^x*cos(x))/d(x)=6*e^x*(cos(x)−sin(x))

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