Derivatives using tables Let h(x)=f(g(x))h(x)=f(g(x)) and p(x)=g(f(x))p(x)=g(f(x)). Use the table to compute the following derivatives. a. h′$$(3)h^{\prime}$$\left(3\right)

Ch. 3 - Derivatives

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

Ch. 3 - Derivatives

Problem 3.7.25a

Chapter 3, Problem 3.7.25a

Derivatives using tables Let $h (x) = f (g (x))$ and $p (x) = g (f (x))$ . Use the table to compute the following derivatives.
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a. $h prime of 3$

Verified step by step guidance

Step 1: Recognize that the function h(x) = f(g(x)) is a composition of functions, and to find its derivative h'(x), we need to use the chain rule.

Step 2: The chain rule states that if h(x) = f(g(x)), then h'(x) = f'(g(x)) * g'(x).

Step 3: To find h'(3), substitute x = 3 into the expression for h'(x), giving us h'(3) = f'(g(3)) * g'(3).

Step 4: Use the table to find the values of g(3), f'(g(3)), and g'(3). First, find g(3) from the table.

Step 5: Once you have g(3), use the table to find f'(g(3)) and g'(3), then multiply these values to find h'(3).

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Key Concepts

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

Chain Rule

The Chain Rule is a fundamental theorem in calculus used to differentiate composite functions. It states that if a function h(x) is composed of two functions f and g, such that h(x) = f(g(x)), then the derivative h'(x) can be found using the formula h'(x) = f'(g(x)) * g'(x). This rule is essential for calculating derivatives of functions that are nested within one another.

Derivative Notation

Derivative notation, such as h'(x) or f'(x), represents the rate of change of a function with respect to its variable. It indicates how the function's output changes as its input changes. Understanding this notation is crucial for interpreting and calculating derivatives, especially when dealing with multiple functions and their compositions.

Function Composition

Function composition occurs when one function is applied to the result of another function. For example, if h(x) = f(g(x)), then g(x) is evaluated first, and its output is used as the input for f. This concept is vital for understanding how to differentiate composite functions and apply the Chain Rule effectively.

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Textbook Question

Derivatives and tangent lines

a. For the following functions and values of a, find f′(a).

f(x) = 8x; a = −3

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Textbook Question

Use definition (1) (p. 133) to find the slope of the line tangent to the graph of f at P.

f(x) = -3x² - 5x + 1; P(1,-7)

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Textbook Question

{Use of Tech} Angle of elevation A small plane, moving at 70 m/s, flies horizontally on a line 400 meters directly above an observer. Let θ be the angle of elevation of the plane (see figure). <IMAGE>

a. What is the rate of change of the angle of elevation dθ/dx when the plane is x=500 m past the observer?

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Textbook Question

60–62. {Use of Tech} Multiple tangent lines Complete the following steps. <IMAGE>

a. Find equations of all lines tangent to the curve at the given value of x.

x+y³−y=1; x=1

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Textbook Question

45–50. Tangent lines Carry out the following steps. <IMAGE>

a. Verify that the given point lies on the curve.