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

Problem

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

Solution

1. Identify the rule needed for the derivative of a product of two functions, u(x)=sin(x) and v(x)=cos(x)

2. Apply the product rule, which states that (d(u)*v)/d(x)=ud(v)/d(x)+vd(u)/d(x)

3. Differentiate the individual trigonometric functions: d(sin(x))/d(x)=cos(x) and d(cos(x))/d(x)=−sin(x)

4. Substitute these derivatives back into the product rule formula:

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

5. Simplify the expression by multiplying the terms:

(d(sin(x))*cos(x))/d(x)=−sin2(x)+cos2(x)

6. Apply the double angle identity for cosine, where cos(2*x)=cos2(x)−sin2(x) to reach the final form.

Final Answer

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

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