Find the Derivative - d/dx 5e^xcos(x)

Problem

d()/d(x)*5*ex*cos(x)

Solution

1. Identify the constant and the product of two functions. The constant 5 can be moved outside the derivative.

d()/d(x)*5*ex*cos(x)=5(d(ex)*cos(x))/d(x)

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

5(d(ex)*cos(x))/d(x)=5*(exd(cos(x))/d(x)+cos(x)d(ex)/d(x))

3. Differentiate the individual functions. The derivative of cos(x) is −sin(x) and the derivative of ex is ex

5*(ex*(−sin(x))+cos(x)*ex)

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

5*ex*(cos(x)−sin(x))

Final Answer

d()/d(x)*5*ex*cos(x)=5*ex*(cos(x)−sin(x))

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