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Ch. 3 - Derivatives
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 3 - DerivativesProblem 3.40a
Chapter 3, Problem 3.40a

Derivatives and tangent lines
a. For the following functions and values of a, find f′(a).
f(x) = √3x; a= 12

Verified step by step guidance
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Step 1: Identify the function f(x) = \(\sqrt{3x}\). This is a composition of functions, where the outer function is the square root and the inner function is 3x.
Step 2: To find the derivative f'(x), use the chain rule. The chain rule states that if you have a composite function f(g(x)), the derivative is f'(g(x)) * g'(x).
Step 3: Differentiate the outer function \(\sqrt{u}\) with respect to u, which is \(\frac{1}{2\sqrt{u}\)}. Here, u = 3x.
Step 4: Differentiate the inner function 3x with respect to x, which is 3.
Step 5: Apply the chain rule: f'(x) = \(\frac{1}{2\sqrt{3x}\)} * 3. Simplify this expression to find f'(x), and then evaluate it at a = 12 to find f'(12).

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Key Concepts

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

Derivatives

A derivative represents the rate of change of a function with respect to its variable. It is defined as the limit of the average rate of change of the function as the interval approaches zero. In practical terms, the derivative at a point gives the slope of the tangent line to the function at that point.
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Derivatives

Tangent Lines

A tangent line to a curve at a given point is a straight line that touches the curve at that point without crossing it. The slope of the tangent line is equal to the derivative of the function at that point. This concept is crucial for understanding how functions behave locally around specific values.
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Function Evaluation

Function evaluation involves substituting a specific value into a function to determine its output. In the context of derivatives, evaluating the function at a point helps in calculating the derivative at that point. For example, to find f′(a), one must first evaluate the function f(x) at x = a.
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Related Practice
Textbook Question

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a. Estimate L' (1.5) and state the physical meaning of this quantity.

Textbook Question

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a. Use implicit differentiation to find dy/dx.

cos y = x; (0, π/2)

Textbook Question

Use definition (2) (p. 135) to find the slope of the line tangent to the graph of f at P.

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Textbook Question

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73. {Use of Tech} Graph the following functions and determine the location of the vertical tangent lines.

a. f(x) = (x-2)^1/3

Textbook Question

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a. Differentiate both sides of the identity cos 2t = cos² t−sin² t to prove that sin 2 t= 2 sin t cos t.

Textbook Question

21–30. Derivatives

a. Use limits to find the derivative function f' for the following functions f.

f(t) = 3t⁴; a= -2, 2

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