Find the Derivative - d/dx 9e^xcos(x)

Problem

d()/d(x)*9*ex*cos(x)

Solution

1. Identify the constant and the functions involved. The constant 9 can be moved outside the derivative, and the remaining expression ex*cos(x) is a product of two functions.

2. Apply the product rule, which states that d()/d(x)*ƒ(x)*g(x)=ƒ(x)d(g(x))/d(x)+g(x)d(ƒ(x))/d(x) Let ƒ(x)=ex and g(x)=cos(x)

3. Differentiate the individual components. The derivative d(ex)/d(x)=ex and the derivative d(cos(x))/d(x)=−sin(x)

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

9*(ex*(−sin(x))+cos(x)*ex)

5. Simplify the expression by factoring out the common term ex and distributing the constant 9

9*ex*(cos(x)−sin(x))

Final Answer

d()/d(x)*9*ex*cos(x)=9*ex*(cos(x)−sin(x))

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