Find the Derivative - d/dx sin(4x)^3

Problem

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

Solution

1. Identify the outer function and the inner function to apply the Power Rule. The expression is of the form u3 where u=sin(4*x)

2. Apply the Power Rule by bringing the exponent to the front and decreasing the power by one.

d(sin(4*x))/d(x)=3*sin(4*x)⋅d(sin(4*x))/d(x)

3. Apply the Chain Rule to the trigonometric function sin(4*x) The derivative of sin(u) is cos(u)⋅d(u)/d(x)

d(sin(4*x))/d(x)=3*sin(4*x)⋅cos(4*x)⋅(d(4)*x)/d(x)

4. Differentiate the innermost linear function 4*x which results in 4

d(sin(4*x))/d(x)=3*sin(4*x)⋅cos(4*x)⋅4

5. Simplify the expression by multiplying the constants 3 and 4

d(sin(4*x))/d(x)=12*sin(4*x)*cos(4*x)

Final Answer

d(sin(4*x))/d(x)=12*sin2(4*x)*cos(4*x)

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