Find the Derivative - d/dx xcos(x)+sin(x)

Problem

d()/d(x)*(x*cos(x)+sin(x))

Solution

1. Apply the sum rule for differentiation to separate the terms.

d()/d(x)*(x*cos(x)+sin(x))=(d(x)*cos(x))/d(x)+d(sin(x))/d(x)

2. Apply the product rule to the first term, where (d(u)*v)/d(x)=ud(v)/d(x)+vd(u)/d(x)

(d(x)*cos(x))/d(x)=xd(cos(x))/d(x)+cos(x)d(x)/d(x)

3. Differentiate the individual trigonometric and power functions.

d(cos(x))/d(x)=−sin(x)

d(x)/d(x)=1

d(sin(x))/d(x)=cos(x)

4. Substitute these derivatives back into the expression.

d()/d(x)*(x*cos(x)+sin(x))=(x*(−sin(x))+cos(1))+cos(x)

5. Simplify the resulting expression by combining like terms.

−x*sin(x)+cos(x)+cos(x)=−x*sin(x)+2*cos(x)

Final Answer

d()/d(x)*(x*cos(x)+sin(x))=2*cos(x)−x*sin(x)

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