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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 3.57c
Chapter 3, Problem 3.57c

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

Verified step by step guidance
1
Identify the function combination for which you need to find the derivative. In this case, it is the derivative of the function f(x) with respect to x, evaluated at x = 1.
Recall that the derivative of a function at a point gives the rate of change of the function at that point. Here, you need to find f'(x) at x = 1.
From the given data, locate the value of f'(x) at x = 1. According to the table, f'(1) = 1/2.
Since the problem asks for the derivative of f(x) at x = 1, and you have already identified f'(1) = 1/2, this is the value you need.
Thus, the first derivative of f(x) at x = 1 is 1/2, which represents the slope of the tangent line to the curve of f(x) at that point.

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Key Concepts

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

Derivative of a Function

The derivative of a function measures how the function's output changes as its input changes. It is defined as the limit of the average rate of change of the function over an interval as the interval approaches zero. In this context, knowing the derivatives of functions ƒ(x) and g(x) at specific points is crucial for finding the derivatives of their combinations.
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Derivatives of Other Trig Functions

Sum Rule of Derivatives

The sum rule states that the derivative of the sum of two functions is equal to the sum of their derivatives. Mathematically, if h(x) = ƒ(x) + g(x), then h'(x) = ƒ'(x) + g'(x). This rule is essential for solving the given problem, as it allows us to find the derivative of the combination ƒ(x) + 1 by simply using the derivative of ƒ(x) since the derivative of a constant (1) is zero.
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Algebra Rules for Finite Sums

Evaluating Derivatives at Specific Points

Evaluating a derivative at a specific point involves substituting the value of x into the derivative function. In this case, we need to find the derivative of the combination at x = 1. This requires using the provided values of the derivatives at that point to compute the final result accurately.
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Critical Points
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

Computer Explorations


Use a CAS to perform the following steps in Exercises 55–62.


b. Using implicit differentiation, find a formula for the derivative dy/dx and evaluate it at the given point P.


2y² + (xy)¹/³ = x² + 2, P(1,1)

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.


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

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

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

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.