Find f′(x), f′′(x), and f′′′(x) for the following functions.f(x) = (x2 - 7x - 8) / (x + 1)

Ch. 3 - Derivatives

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

Ch. 3 - Derivatives

Problem 71

Chapter 3, Problem 71

Find f′(x), f′′(x), and f′′′(x) for the following functions.
f(x) = (x² - 7x - 8) / (x + 1)

Verified step by step guidance

Step 1: Identify the function f(x) = $\frac{x^2 - 7x - 8}{x + 1}$. This is a rational function, so we will use the quotient rule to find the derivatives.

Step 2: Recall the quotient rule for derivatives: if you have a function $\frac{u(x)}{v(x)}$, then its derivative is given by $\frac{d}{dx}\[\left$($\frac{u}{v}\]\right$) = $\frac{u'v - uv'}{v^2}$. Here, u(x) = x^2 - 7x - 8 and v(x) = x + 1.

Step 3: Compute the first derivative f'(x) using the quotient rule. First, find the derivatives of the numerator and denominator: u'(x) = 2x - 7 and v'(x) = 1. Then apply the quotient rule: f'(x) = $\frac{(2x - 7)(x + 1) - (x^2 - 7x - 8)(1)}{(x + 1)^2}$.

Step 4: Simplify the expression for f'(x) by expanding and combining like terms in the numerator. This will give you a simplified form of the first derivative.

Step 5: To find the second derivative f''(x), differentiate f'(x) using the quotient rule again. Finally, differentiate f''(x) to find the third derivative f'''(x), ensuring to simplify at each step.

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Key Concepts

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

Differentiation

Differentiation is the process of finding the derivative of a function, which represents the rate of change of the function with respect to its variable. It is a fundamental concept in calculus that allows us to analyze the behavior of functions, including their slopes and rates of change. The first derivative, f′(x), gives the slope of the tangent line to the curve at any point x.

Higher-Order Derivatives

Higher-order derivatives are derivatives of derivatives. The second derivative, f′′(x), provides information about the curvature of the function, indicating whether the function is concave up or down. The third derivative, f′′′(x), can give insights into the rate of change of the curvature, which is useful in understanding the function's behavior in more detail.

Quotient Rule

The Quotient Rule is a formula used to differentiate functions that are expressed as the quotient of two other functions. If f(x) = g(x)/h(x), the derivative is given by f′(x) = (g′(x)h(x) - g(x)h′(x)) / (h(x))². This rule is essential for the given function, as it involves dividing a polynomial by another polynomial, requiring careful application of this rule to find the derivatives accurately.

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Textbook Question

The right-sided and left-sided derivatives of a function at a point $a$ are given by $f_{+}^{'} (a) = \lim_{h \to 0^{+}} \frac{f (a + h) - f (a)}{h}$ and $f_{-}^{'} (a) = \lim_{h \to 0^{-}} \frac{f (a + h) - f (a)}{h}$ , respectively, provided these limits exist. The derivative $f^{'} (a)$ exists if and only if $f_{+}^{'} (a) = f_{-}^{'} (a)$ .

Compute $f_{+}^{'} (a)$ and $f_{-}^{'} (a)$ at the given point $a$ .

$f (x) = ∣ x - 2 ∣$ ; $a equals 2$

Textbook Question

The following equations implicitly define one or more functions.

c. Use the functions found in part (b) to graph the given equation.

y² = x²(4 − x) / 4 + x (right strophoid)

Textbook Question

The following equations implicitly define one or more functions.

a. Find dy/dx using implicit differentiation.

y² = x²(4 − x) / 4 + x (right strophoid)

Textbook Question

The right-sided and left-sided derivatives of a function at a point $a$ are given by $f_{+}^{'} (a) = \lim_{h \to 0^{+}} \frac{f (a + h) - f (a)}{h}$ and $f_{-}^{'} (a) = \lim_{h \to 0^{-}} \frac{f (a + h) - f (a)}{h}$ , respectively, provided these limits exist. The derivative $f^{$\prime$}$\left$(a$\right$)$ exists if and only if $f_{+}^{$\prime$}$\left$(a$\right$)=f_{-}^{$\prime$}$\left$(a$\right$)$ .

Compute $f_{+}^{$\prime$}$\left$(a$\right$)$ and $f_{-}^{$\prime$}$\left$(a$\right$)$ at the given point $a$ .

$f (x) = {\begin{matrix} 4 - x^{2} if x \leq 1 \\ 2 x + 1 if x > 1 \end{matrix}$ ; $a equals 1$

Textbook Question

The following equations implicitly define one or more functions.

b. Solve the given equation for y to identify the implicitly defined functions y=f₁(x), y = f₂(x), ….

y² = x²(4 − x) / 4 + x (right strophoid)

Textbook Question

First and second derivatives Find f′(x),f′′(x).

f(x) = x/x+2