Find the Derivative - d/dx cos(x)sin(x)

Problem

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

Solution

1. Identify the rule needed for the derivative of a product of two functions, which is the product rule: (d(u)*v)/d(x)=ud(v)/d(x)+vd(u)/d(x)

2. Assign the functions where u=cos(x) and v=sin(x)

3. Differentiate each part individually: d(cos(x))/d(x)=−sin(x) and d(sin(x))/d(x)=cos(x)

4. Substitute these into the product rule formula: cos(x)*cos(x)+sin(x)*(−sin(x))

5. Simplify the expression using trigonometric notation: cos2(x)−sin2(x)

6. Apply the double angle identity for cosine: cos(2*x)=cos2(x)−sin2(x)

Final Answer

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

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