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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 3.6.76c
Chapter 3, Problem 3.6.76c

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.


c. f(x) / (g(x) + 1), x = 1

Verified step by step guidance
1
Step 1: Recognize that the function to differentiate is f(x) / (g(x) + 1). To find its derivative, use the quotient rule: (u/v)' = (u'v - uv') / v^2, where u = f(x) and v = g(x) + 1.
Step 2: Compute u' and v'. From the table, u = f(x) and u' = f'(x). Similarly, v = g(x) + 1, so v' = g'(x). Use the values at x = 1: f'(x) = -1/3 and g'(x) = -8/3.
Step 3: Substitute the values of u, u', v, and v' into the quotient rule formula. At x = 1, u = f(x) = 3, v = g(x) + 1 = -4 + 1 = -3, u' = -1/3, and v' = -8/3.
Step 4: Plug these values into the formula: (u/v)' = [(u' * v) - (u * v')]/v^2. This becomes [(-1/3 * -3) - (3 * -8/3)] / (-3)^2.
Step 5: Simplify the numerator and denominator separately. The numerator involves multiplication and subtraction, while the denominator is the square of v. Combine these results to express the derivative.

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Key Concepts

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

Quotient Rule

The quotient rule is used to find the derivative of a function that is the quotient of two differentiable functions. If you have a function h(x) = f(x)/g(x), the derivative h'(x) is given by [g(x)f'(x) - f(x)g'(x)] / [g(x)]^2. This rule is essential for differentiating the given function f(x) / (g(x) + 1).
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The Quotient Rule

Derivative Evaluation

Derivative evaluation involves substituting specific values into the derivative formula to find the rate of change at a particular point. In this problem, you need to evaluate the derivative of the function at x = 1 using the values provided in the table for f(x), g(x), f'(x), and g'(x). This step is crucial for obtaining the final numerical result.
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Evaluate Logarithms

Function Composition

Function composition involves combining two functions to form a new function, such as f(x) / (g(x) + 1). Understanding how to manipulate and differentiate composed functions is key to solving this problem, as it requires applying the quotient rule to a function that includes a sum in the denominator.
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Related Practice
Textbook Question

Motion Along a Coordinate Line


Exercises 1–6 give the positions s = f(t) of a body moving on a coordinate line, with s in meters and t in seconds.


c. When, if ever, during the interval does the body change direction?


s = 25/(t + 5), −4 ≤ t ≤ 0

Textbook Question

Right circular cylinder The total surface area S of a right circular cylinder is related to the base radius r and height h by the equation S = 2πr² + 2πrh.


c. How is dS/dt related to dr/dt and dh/dt if neither r nor h is constant?

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.


c. ƒ(x) , x = 1

g(x) + 1

Textbook Question

Diagonals If x, y, and z are lengths of the edges of a rectangular box, then the common length of the box’s diagonals is s = √(x² + y² + z²).

c. How are dx/dt, dy/dt, and dz/dt related if s is constant?

Textbook Question

Differentiability and Continuity on an Interval


Each figure in Exercises 45–50 shows the graph of a function over a closed interval D. At what domain points does the function appear to be


c. neither continuous nor differentiable?


Give reasons for your answers.


Textbook Question

Particle motion At time t, the position of a body moving along the s-axis is s = t³ − 6t² + 9t m.


c. Find the total distance traveled by the body from t = 0 to t = 2.