Find the derivative of the function f(x) = x^3(1-2x) .

Question:

Find the derivative of the function f(x) = {eq}x^3(1-2x) {/eq}.

Power Rule for Differentiation:

This function can be derived utilizing the product rule for differentiation; however, distributing the {eq}x^{3} {/eq} and then utilizing the power rule to differentiate would be easier.

For {eq}f(x)=x^{n} {/eq} where {eq}n \neq 0 {/eq}, {eq}f'(x)=nx^{n-1} {/eq}.

Answer and Explanation:

Step 1. Distribute {eq}x^{3} {/eq}.

{eq}\begin{align} \displaystyle f(x)\displaystyle &=x^{3}\left(1-2x\right)\\ \displaystyle &=x^{3}-2x^{4}\\ \end{align} {/eq}

Step 2. Utilize the power rule to differentiate.

{eq}\begin{align} \displaystyle f(x)\displaystyle &=x^{3}-2x^{4}\\ \displaystyle f'(x)\displaystyle &=3x^{2}-8x^{3}\\ \end{align} {/eq}


Learn more about this topic:

Power Rule for Derivatives: Examples & Explanation

from High School Precalculus: Help and Review

Chapter 19 / Lesson 18
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