Find the Derivative - d/dx 4sin(x)cos(x)

Problem

d()/d(x)*4*sin(x)*cos(x)

Solution

1. Identify the expression as a product of trigonometric functions and recognize that the constant multiple rule allows us to move the 4 outside the derivative.

d()/d(x)*4*sin(x)*cos(x)=4d()/d(x)*sin(x)*cos(x)

2. Apply the double-angle identity for sine, which states sin(2*x)=2*sin(x)*cos(x) to simplify the expression before differentiating.

4*sin(x)*cos(x)=2*(2*sin(x)*cos(x))

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

3. Differentiate the simplified expression 2*sin(2*x) using the chain rule, where the derivative of sin(u) is cos(u)⋅d(u)/d(x)

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

4. Calculate the derivative of the inner function 2*x which is 2, and multiply it by the outer derivative.

2*cos(2*x)⋅2=4*cos(2*x)

Final Answer

d()/d(x)*4*sin(x)*cos(x)=4*cos(2*x)

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