Derivatives Find and simplify the derivative of the following functions.
f(x) = x /x+1

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Step 1: Recognize that the function $ f(x) = \frac{x}{x+1} $ is a quotient of two functions, $ u(x) = x $ and $ v(x) = x+1 $. To find the derivative, we will use the Quotient Rule.

Step 2: Recall the Quotient Rule, which states that if $ f(x) = \frac{u(x)}{v(x)} $, then $ f'(x) = \frac{u'(x)v(x) - u(x)v'(x)}{(v(x))^2} $.

Step 3: Differentiate $ u(x) = x $ to get $ u'(x) = 1 $, and differentiate $ v(x) = x+1 $ to get $ v'(x) = 1 $.

Step 4: Substitute $ u(x) $, $ u'(x) $, $ v(x) $, and $ v'(x) $ into the Quotient Rule formula: $ f'(x) = \frac{1 \cdot (x+1) - x \cdot 1}{(x+1)^2} $.

Step 5: Simplify the expression obtained in Step 4 to find the simplified form of the derivative.

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Derivatives

A derivative represents the rate of change of a function with respect to its variable. It is a fundamental concept in calculus that provides information about the slope of the tangent line to the curve of the function at any given point. The derivative can be computed using various rules, such as the power rule, product rule, quotient rule, and chain rule.

Quotient Rule

The quotient rule is a method for finding the derivative of a function that is the ratio of two other functions. If you have a function f(x) = g(x)/h(x), the derivative f'(x) is given by (g'(x)h(x) - g(x)h'(x)) / (h(x))^2. This rule is essential when differentiating functions that are expressed as fractions.

Simplification of Derivatives

After finding the derivative of a function, simplification is often necessary to express the result in its simplest form. This may involve factoring, reducing fractions, or combining like terms. Simplifying the derivative can make it easier to analyze the function's behavior, such as identifying critical points and determining concavity.

관련 실천

교과서 질문

Using identities Use the identity sin 2x=2 sin x cos x sin 2 to find d/dx (sin 2x). Then use the identity cos 2x = cos² x−sin² x to express the derivative of sin 2x in terms of cos 2x.

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교과서 질문

Orthogonal trajectories Two curves are orthogonal to each other if their tangent lines are perpendicular at each point of intersection (recall that two lines are perpendicular to each other if their slopes are negative reciprocals). A family of curves forms orthogonal trajectories with another family of curves if each curve in one family is orthogonal to each curve in the other family. For example, the parabolas y = cx² form orthogonal trajectories with the family of ellipses x²+2y² = k, where c and k are constants (see figure).

Find dy/dx for each equation of the following pairs. Use the derivatives to explain why the families of curves form orthogonal trajectories. <IMAGE>

y = cx²; x²+2y² = k, where c and k are constants

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교과서 질문

15–48. Derivatives Find the derivative of the following functions.

y = 10^x(In 10^x-1)

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교과서 질문

27–76. Calculate the derivative of the following functions.

$y = {(x^{2} + 2 x + 7)}^{8}$