Find the Derivative - d/dx x(1-2x)^3

Problem

d()/d(x)*x*(1−2*x)3

Solution

1. Identify the rule needed for the expression, which is a product of two functions: u=x and v=(1−2*x)3

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

3. Differentiate the first part, u=x which gives d(x)/d(x)=1

4. Differentiate the second part, v=(1−2*x)3 using the chain rule.

5. Apply the chain rule to find d(1−2*x)/d(x)=3*(1−2*x)2⋅d(1−2*x)/d(x)

6. Calculate the inner derivative d(1−2*x)/d(x)=−2

7. Combine the chain rule results to get d(v)/d(x)=3*(1−2*x)2*(−2)=−6*(1−2*x)2

8. Substitute these components back into the product rule formula.

x*(−6*(1−2*x)2)+(1−2*x)3*(1)

9. Factor out the common term (1−2*x)2 to simplify the expression.

(1−2*x)2*(−6*x+(1−2*x))

10. Combine like terms inside the parentheses.

(1−2*x)2*(1−8*x)

Final Answer

d()/d(x)*x*(1−2*x)3=(1−2*x)2*(1−8*x)

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