Find the Derivative - d/dx 7x^2sin(x)+14xcos(x)-14sin(x)

Problem

d()/d(x)*(7*x2*sin(x)+14*x*cos(x)−14*sin(x))

Solution

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

d()/d(x)*(7*x2*sin(x))+d()/d(x)*(14*x*cos(x))−d()/d(x)*(14*sin(x))

2. Apply the product rule to the first term 7*x2*sin(x) where d()/d(x)*u*v=ud(v)/d(x)+vd(u)/d(x)

(d(7)*x2*sin(x))/d(x)=7*x2*cos(x)+14*x*sin(x)

3. Apply the product rule to the second term 14*x*cos(x)

(d(14)*x*cos(x))/d(x)=14*x*(−sin(x))+14*cos(x)

(d(14)*x*cos(x))/d(x)=−14*x*sin(x)+14*cos(x)

4. Differentiate the third term using the basic derivative rule for sine.

(d(14)*sin(x))/d(x)=14*cos(x)

5. Combine the results from all terms and simplify by canceling out terms that sum to zero.

(7*x2*cos(x)+14*x*sin(x))+(−14*x*sin(x)+14*cos(x))−(14*cos(x))

7*x2*cos(x)+14*x*sin(x)−14*x*sin(x)+14*cos(x)−14*cos(x)

7*x2*cos(x)

Final Answer

d()/d(x)*(7*x2*sin(x)+14*x*cos(x)−14*sin(x))=7*x2*cos(x)

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