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

Problem

(d(6)*ex*cos(x))/d(x)

Solution

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

(d(6)*ex*cos(x))/d(x)=6(d(ex)*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=ex and v=cos(x)

6(d(ex)*cos(x))/d(x)=6*(exd(cos(x))/d(x)+cos(x)d(ex)/d(x))

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

6*(ex*(−sin(x))+cos(x)*ex)

4. Simplify the expression by factoring out the common term ex

6*ex*(cos(x)−sin(x))

Final Answer

(d(6)*ex*cos(x))/d(x)=6*ex*(cos(x)−sin(x))

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