Find dy/dx y=sin(x)cos(x)

Problem

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

Solution

1. Identify the rule needed for differentiation. Since the expression is a product of two functions, sin(x) and cos(x) use the product rule: (d(u)*v)/d(x)=ud(v)/d(x)+vd(u)/d(x)

2. Assign the variables for the product rule. Let u=sin(x) and v=cos(x)

3. Differentiate each part individually. The derivative of sin(x) is cos(x) and the derivative of cos(x) is −sin(x)

4. Apply the product rule formula by substituting the functions and their derivatives.

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

5. Simplify the resulting expression using trigonometric notation.

d(y)/d(x)=cos2(x)−sin2(x)

6. Recognize the trigonometric identity. The expression cos2(x)−sin2(x) is equivalent to the double-angle identity cos(2*x)

Final Answer

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

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