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Ch. 3 - Derivatives
Hass - Thomas' Calculus 15th Edition
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Tutti i libri di testoHass 15th EditionCh. 3 - DerivativesProblema 3.6.76c
Capitolo 3, Problema 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

Guida verificata passo dopo passo
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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Concetti chiave

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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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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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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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