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

Problem

d()/d(x)*x*(1−2*x)3

Solution

1. Identify the rule needed for the expression, which is the product rule d()/d(x)*u*v=ud(v)/d(x)+vd(u)/d(x) where u=x and v=(1−2*x)3

2. Differentiate the first part u=x with respect to x

d(x)/d(x)=1

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

d(1−2*x)/d(x)=3*(1−2*x)2⋅d(1−2*x)/d(x)

d(1−2*x)/d(x)=3*(1−2*x)2⋅(−2)

d(1−2*x)/d(x)=−6*(1−2*x)2

4. Apply the product rule formula by combining the results.

(d(x)*(1−2*x)3)/d(x)=x*(−6*(1−2*x)2)+(1−2*x)3*(1)

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

(d(x)*(1−2*x)3)/d(x)=(1−2*x)2*(−6*x+(1−2*x))

6. Combine the terms inside the parentheses.

(d(x)*(1−2*x)3)/d(x)=(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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