Find the Derivative - d/dx x^2sin(x)+2xcos(x)-2sin(x)

Problem

d()/d(x)*(x2*sin(x)+2*x*cos(x)−2*sin(x))

Solution

1. Apply the sum rule to differentiate each term of the expression individually.

d()/d(x)*(x2*sin(x))+d()/d(x)*(2*x*cos(x))−d()/d(x)*(2*sin(x))

2. Apply the product rule to the first term x2*sin(x) where d()/d(x)*u*v=ud(v)/d(x)+vd(u)/d(x)

(d(x2)*sin(x))/d(x)=x2*cos(x)+2*x*sin(x)

3. Apply the product rule to the second term 2*x*cos(x)

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

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

4. Differentiate the third term using the constant multiple rule.

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

5. Combine the results from all terms into a single expression.

(x2*cos(x)+2*x*sin(x))+(−2*x*sin(x)+2*cos(x))−(2*cos(x))

6. Simplify the expression by canceling out terms with opposite signs.

x2*cos(x)+2*x*sin(x)−2*x*sin(x)+2*cos(x)−2*cos(x)

x2*cos(x)

Final Answer

d()/d(x)*(x2*sin(x)+2*x*cos(x)−2*sin(x))=x2*cos(x)

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