Find the Derivative - d/dx cos((1-2x)^2)

Problem

d(cos((1−2*x)2))/d(x)

Solution

1. Identify the outer function and the inner function to apply the chain rule. The outer function is cos(u) where u=(1−2*x)2

2. Differentiate the outer function with respect to u The derivative of cos(u) is −sin(u)

d(cos(u))/d(u)=−sin(u)

3. Apply the chain rule by multiplying by the derivative of the inner function u=(1−2*x)2

d(cos((1−2*x)2))/d(x)=−sin((1−2*x)2)⋅d(1−2*x)/d(x)

4. Differentiate the inner function (1−2*x)2 using the chain rule again. The derivative of v2 is 2*v where v=1−2*x

d(1−2*x)/d(x)=2*(1−2*x)⋅d(1−2*x)/d(x)

5. Calculate the derivative of the innermost part 1−2*x which is −2

d(1−2*x)/d(x)=−2

6. Combine all the components together.

d(cos((1−2*x)2))/d(x)=−sin((1−2*x)2)⋅2*(1−2*x)⋅(−2)

7. Simplify the expression by multiplying the constants.

−1⋅2⋅−2=4

Final Answer

d(cos((1−2*x)2))/d(x)=4*(1−2*x)*sin((1−2*x)2)

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