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

Problem

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

Solution

1. Identify the rule needed for the derivative of a product of two functions, u(x)=ex and v(x)=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: d(ex)/d(x)=ex and d(cos(x))/d(x)=−sin(x)

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

5. Factor out the common term ex to simplify the expression.

Final Answer

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

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