Question

Use the Product Rule to show that $\left(f^{2}\right)^{\prime}=2 f f^{\prime}$

   Use the Product Rule to show that $\left(f^{2}\right)^{\prime}=2 f f^{\prime}$

Calculus for AP

Calculus for AP

Jon Rogawski & Ray… 2nd Edition

Chapter 3, Problem 55

Instant Answer

Solved by Expert Alex Johnson

02/10/2020

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Step 1: We start by expressing $f^2$ as $f \cdot f$. Show more…

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Use the Product Rule to show that $\left(f^{2}\right)^{\prime}=2 f f^{\prime}$

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Key Concepts

Differentiation of Powers of a Function

When a function is raised to a power, such as f(x)^2, it can be understood as the product of the function with itself. Applying the product rule in this context—treating f(x)^2 as f(x) multiplied by f(x)—enables you to derive the relationship between the function and its derivative. This approach is crucial for verifying results like (f^2)' = 2f f'.

Product Rule

The product rule is a key concept in differential calculus that deals with the differentiation of the product of two functions. According to this rule, if you have two functions, u(x) and v(x), their derivative with respect to x is given by u'(x)v(x) + u(x)v'(x). This principle allows one to break down complex products into simpler parts that can be individually differentiated.

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