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Ch. 3 - Derivatives
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 3 - DerivativesProblem 3.6.75g
Chapter 3, Problem 3.6.75g

Suppose that functions f and g and their derivatives with respect to x have the following values at x = 2 and x = 3.


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Find the derivatives with respect to x of the following combinations at the given value of x.


g. 1 / g²(x), x = 3

Verified step by step guidance
1
Step 1: Recall the derivative rule for the reciprocal of a function. If h(x) = 1 / g²(x), then h'(x) can be found using the chain rule and the power rule. Specifically, h'(x) = -2 * g(x)^(-3) * g'(x).
Step 2: Substitute the given value of x = 3 into the formula. From the table, g(x) = -4 and g'(x) = 5 at x = 3.
Step 3: Replace g(x) and g'(x) in the derivative formula. This gives h'(x) = -2 * (-4)^(-3) * 5.
Step 4: Simplify the expression step by step. First, calculate (-4)^(-3), which is the reciprocal of (-4)^3. Then multiply by -2 and g'(x).
Step 5: The simplified expression will give the derivative h'(x) at x = 3. Ensure all calculations are consistent with the rules of exponents and multiplication.

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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 differentiation technique used when dealing with composite functions. It states that the derivative of a composite function f(g(x)) is f'(g(x)) * g'(x). In this problem, the chain rule helps differentiate expressions involving powers of functions, such as g²(x), by considering the derivative of the outer function and multiplying it by the derivative of the inner function.
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Intro to the Chain Rule

Power Rule

The power rule is a basic differentiation rule that states if f(x) = x^n, then f'(x) = n*x^(n-1). This rule is essential for finding the derivative of functions raised to a power, such as g²(x). In the context of the problem, it helps simplify the differentiation process by providing a straightforward method to handle powers of functions.
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Power Rules

Quotient Rule

The quotient rule is used to differentiate functions that are ratios of two other functions. It states that if h(x) = f(x)/g(x), then h'(x) = (f'(x)g(x) - f(x)g'(x))/g²(x). In this problem, the quotient rule is applied to find the derivative of 1/g²(x), which involves differentiating a function in the form of a reciprocal, requiring careful application of the rule to ensure accuracy.
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Related Practice
Textbook Question

Analyzing Motion Using Graphs


[Technology Exercise] Exercises 31–34 give the position function s = f(t) of an object moving along the s-axis as a function of time t. Graph f together with the velocity function v(t) = ds/dt = f'(t) and the acceleration function a(t) = d²s/dt² = f''(t). Comment on the object’s behavior in relation to the signs and values of v and a. Include in your commentary such topics as the following:


f. When is it farthest from the axis origin?


s = t³ - 6t² + 7t, 0 ≤ t ≤ 4

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Textbook Question

Suppose that the functions f and g and their derivatives with respect to x have the following values at x = 0 and x = 1.


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Find the derivatives with respect to x of the following combinations at the given value of x.


g. f(x + g(x)), x = 0

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Textbook Question

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In Exercises 13–16, differentiate the functions and find the slope of the tangent line at the given value of the independent variable.


f(x) = x + 9/x, x = −3

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Textbook Question

Finding Derivative Functions and Values 


Using the definition, calculate the derivatives of the functions in Exercises 1–6. Then find the values of the derivatives as specified.


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Textbook Question

Suppose that functions f and g and their derivatives with respect to x have the following values at x = 2 and x = 3.


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Find the derivatives with respect to x of the following combinations at the given value of x.


h. √(f²(x) + g²(x)), x = 2

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Textbook Question

Suppose that functions ƒ(x) and g(x) and their first derivatives have the following values at x = 0 and x = 1.


x ƒ(x) g(x) ƒ'(x) g'(x)

0 1 1 -3 1/2

1 3 5 1/2 -4


Find the first derivatives of the following combinations at the given value of x.


g. ƒ(x + g(x)), x = 0

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