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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 3.37a
Chapter 3, Problem 3.37a

Derivatives and tangent lines
a. For the following functions and values of a, find f′(a).
f(x) = 1/ √x; a= 1/4

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1
Step 1: Identify the function and the point of interest. The function given is \( f(x) = \frac{1}{\sqrt{x}} \) and we need to find the derivative at \( a = \frac{1}{4} \).
Step 2: Rewrite the function in a form that is easier to differentiate. Express \( f(x) = \frac{1}{\sqrt{x}} \) as \( f(x) = x^{-1/2} \).
Step 3: Use the power rule for differentiation. The power rule states that \( \frac{d}{dx}[x^n] = nx^{n-1} \). Apply this to \( f(x) = x^{-1/2} \) to find \( f'(x) \).
Step 4: Differentiate \( f(x) = x^{-1/2} \) using the power rule. This gives \( f'(x) = -\frac{1}{2}x^{-3/2} \).
Step 5: Evaluate the derivative at \( x = \frac{1}{4} \). Substitute \( x = \frac{1}{4} \) into \( f'(x) = -\frac{1}{2}x^{-3/2} \) to find \( f'(\frac{1}{4}) \).

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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 and is often used in optimization problems.
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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 the point a before applying derivative rules.
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Related Practice
Textbook Question

62–65. {Use of Tech} Graphing f and f'

a. Graph f with a graphing utility.

f(x)=e^−x tan^−1 x on [0,∞)

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Textbook Question

Derivatives of inverse functions from a table Use the following tables to determine the indicated derivatives or state that the derivative cannot be determined. <IMAGE>

a. (f^-1)'(4)

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Textbook Question

Use the definition of the derivative to determine d/dx(ax²+bx+c), where a, b, and c are constants.

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Textbook Question

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

f(x) = 2x + 1; P(0,1)

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Textbook Question

79–82. {Use of Tech} Visualizing tangent and normal lines <IMAGE>

a. Determine an equation of the tangent line and the normal line at the given point (x0, y0) on the following curves. (See instructions for Exercises 73–78.)

(x²+y²)² = 25/3 (x²-y²); (x0,y0) = (2,-1) (lemniscate of Bernoulli)

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Textbook Question

The volume V of a sphere of radius r changes over time t.

a. Find an equation relating dV/dt to dr/dt.

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