Find the Derivative - d/dx y=4e^xcos(x)

Problem

d()/d(x)*4*e^x*cos(x)

Solution

1. Identify the function as a product of two terms, 4*e^x and cos(x) which requires the use of the product rule.

2. Apply the product rule formula, which states that d()/d(x)*[ƒ(x)*g(x)]=ƒ(x)^′*g(x)+ƒ(x)*g(x)^′

3. Differentiate the first term 4*e^x to get 4*e^x

4. Differentiate the second term cos(x) to get −sin(x)

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

(d(4)*e^x*cos(x))/d(x)=(4*e^x)*(cos(x))+(4*e^x)*(−sin(x))

6. Factor out the common term 4*e^x to simplify the expression.

(d(4)*e^x*cos(x))/d(x)=4*e^x*(cos(x)−sin(x))

Final Answer

(d(4)*e^x*cos(x))/d(x)=4*e^x*(cos(x)−sin(x))

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