Ch. 3 - Derivatives

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

Briggs 3rd Edition

Ch. 3 - Derivatives

Problem 3.7.112b

Chapter 3, Problem 3.7.112b

109-112 {Use of Tech} Calculating limits The following limits are the derivatives of a composite function g at a point a.
b. Use the Chain Rule to find each limit. Verify your answer by using a calculator.
$\lim_{h \to 0} \frac{\frac{1}{3 {({(1 + h)}^{5} + 7)}^{10}} - \frac{1}{3 {(8)}^{10}}}{h}$

Verified step by step guidance

Step 1: Recognize that the given limit represents the derivative of a composite function g at a point a. The expression inside the limit is of the form $ \frac{f(x+h) - f(x)}{h} $, which is the definition of the derivative.

Step 2: Identify the inner function and the outer function. Here, the inner function is $ u(h) = (1+h)^5 + 7 $ and the outer function is $ v(u) = \frac{1}{3u^{10}} $.

Step 3: Apply the Chain Rule to find the derivative. The Chain Rule states that $ \frac{d}{dh} v(u(h)) = v'(u(h)) \cdot u'(h) $.

Step 4: Calculate the derivatives: $ u'(h) = \frac{d}{dh}((1+h)^5 + 7) = 5(1+h)^4 $ and $ v'(u) = \frac{d}{du}(\frac{1}{3u^{10}}) = -\frac{10}{3u^{11}} $.

Step 5: Substitute $ u(h) $ and $ u'(h) $ into the Chain Rule expression: $ \frac{d}{dh} v(u(h)) = -\frac{10}{3((1+h)^5 + 7)^{11}} \cdot 5(1+h)^4 $. Evaluate this expression at $ h = 0 $ to find the limit.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Limits

Limits are fundamental in calculus, representing the value that a function approaches as the input approaches a certain point. In this context, the limit as h approaches 0 is crucial for determining the derivative of the function at a specific point. Understanding limits allows us to analyze the behavior of functions near points of interest, which is essential for applying the Chain Rule.

Chain Rule

The Chain Rule is a formula for computing the derivative of a composite function. It states that if you have two functions, f(g(x)), the derivative can be found by multiplying the derivative of the outer function f with the derivative of the inner function g. This rule is particularly useful in the given problem, where the limit involves a composite function, allowing for the simplification of the differentiation process.

Derivatives

Derivatives represent the rate of change of a function with respect to its variable. They provide information about the slope of the tangent line to the function's graph at a given point. In this question, the limit expression is used to find the derivative of a composite function, which is essential for understanding how the function behaves locally around the point of interest.

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