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

Problem

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

Solution

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

2. Apply the Power Rule to the outer function u3

d(u3)/d(u)=3*u2

3. Differentiate the inner function u=cos(4*x) with respect to x which requires another application of the Chain Rule.

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

4. Calculate the derivative of the innermost linear term 4*x

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

5. Combine all parts using the Chain Rule formula.

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

6. Simplify the expression by multiplying the constants.

3⋅(−1)⋅4=−12

Final Answer

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

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