Question

Find $f^{\prime}(x)$. $f(x)=(2 x+1)\left(1+\frac{1}{x}\right)\left(x^{-3}+7\right)$

   Find $f^{\prime}(x)$.
$f(x)=(2 x+1)\left(1+\frac{1}{x}\right)\left(x^{-3}+7\right)$

Calculus Early Transcendentals

Calculus Early Transcendentals

Howard Anton, Irl… 9th Edition

Chapter 2, Problem 17

Instant Answer

Solved by Expert Amy Jiang

08/14/2020

Step 1

Step 1: First, we identify the three functions that are being multiplied together: $u(x) = 2x+1$, $v(x) = 1+\frac{1}{x}$, and $w(x) = x^{-3}+7$. Show more…

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Find $f^{\prime}(x)$. $f(x)=(2 x+1)\left(1+\frac{1}{x}\right)\left(x^{-3}+7\right)$

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Key Concepts

Product Rule

The product rule is a fundamental principle in differentiation used when finding the derivative of the product of two or more functions. It states that the derivative of a product of functions can be obtained by differentiating each function individually while keeping the others constant and summing the results. When applied to a product of three functions, the rule extends by differentiating each function in turn and multiplying by the undifferentiated other functions.

Power Rule

The power rule is a basic technique for differentiating functions of the form x^n, where n is any real number. According to this rule, the derivative of x^n is n times x raised to the power of n-1. This rule is particularly useful when dealing with polynomial terms or any function that can be expressed in terms of a power of x.

Constant Multiple and Sum Rules

The constant multiple rule and the sum (or difference) rule are key differentiation rules that simplify the process of taking derivatives. The constant multiple rule states that a constant multiplied by a function can be taken out of the derivative, and the derivative is therefore the constant times the derivative of the function. Similarly, the sum rule explains that the derivative of a sum of functions is the sum of their derivatives. These rules help break down and simplify the differentiation of more complex algebraic expressions.

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