Find the Derivative - d/dx f(x)=xsin(x)+cos(x)

Problem

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

Solution

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

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

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

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

3. Evaluate the derivatives of the basic trigonometric functions and the power function.

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

d(cos(x))/d(x)=−sin(x)

d(x)/d(x)=1

4. Substitute the derivatives back into the expanded expression.

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

5. Simplify the expression by combining like terms.

x*cos(x)+sin(x)−sin(x)=x*cos(x)

Final Answer

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

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