Find the Derivative - d/dx 8e^xcos(x)

Problem

d()/d(x)*8*e^x*cos(x)

Solution

1. Identify the constant and the functions involved. The expression is a constant 8 multiplied by the product of two functions, e^x and cos(x)

2. Apply the constant multiple rule to move the 8 outside the derivative.

d()/d(x)*8*e^x*cos(x)=8(d(e^x)*cos(x))/d(x)

3. Apply the product rule, which states (d(u)*v)/d(x)=ud(v)/d(x)+vd(u)/d(x) where u=e^x and v=cos(x)

8(d(e^x)*cos(x))/d(x)=8*(e^xd(cos(x))/d(x)+cos(x)d(e^x)/d(x))

4. Differentiate the individual functions. The derivative of cos(x) is −sin(x) and the derivative of e^x is e^x

8*(e^x*(−sin(x))+cos(x)*e^x)

5. Simplify the expression by factoring out the common term e^x and rearranging the terms.

8*e^x*(cos(x)−sin(x))

Final Answer

d()/d(x)*8*e^x*cos(x)=8*e^x*(cos(x)−sin(x))

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