Question

In Exercises, find the higher-order derivative. $$ f^{\prime \prime \prime}(x)=(3 x-1) / x $$

Short answer

The fourth derivative of the function is \( f^{(4)}(x) = 1 / x^2 \)

Step-by-step solution

Identify the u and v

Let's denote \( u = 3x-1, v = x\). Compute the derivatives of \( u \) and \( v \). Thus, \( u' = 3, v' = 1 \)

Apply Quotient Rule

Use the quotient rule of differentiation to find the fourth derivative of the function. This would be \((x*3 - (3x - 1)*1) / x^2\).

Simplification

Simplify the above expression and we get \( (3x - 3x + 1) / x^2 = 1 / x^2 \)

Key concepts

Quotient Rule

The quotient rule is a technique for performing differentiation when dealing with a function that is the division of two other functions. In calculus, it's an essential tool for finding the derivative of ratios of functions. For example, if you have a function that can be expressed as \( \frac{u}{v} \) where \( u \) and \( v \) are both functions of \( x \) that are differentiable, the quotient rule states that the derivative of this function is \( \frac{u'v - uv'}{v^2} \).

To apply the quotient rule effectively, first identify \( u \) and \( v \) as separate functions, then find their derivatives, \( u' \) and \( v' \) respectively. Subsequently, substitute these values into the quotient rule formula for computing the derivative. It's vital to undertake this process methodically to avoid errors and ensure that the derivative is calculated correctly. Understanding the quotient rule is fundamental for finding both first and higher-order derivatives when working with quotients of functions.

Differentiation

Differentiation is a principal operation in calculus that measures how a function changes as its input changes. It's the process of finding a derivative, which represents an instantaneous rate of change. This concept allows us to understand the slope of a curve at any given point, and it is widely used in various disciplines like physics, engineering, and economics.

In practice, there are several rules and techniques for differentiation, including the power rule, product rule, chain rule, and quotient rule. Each rule has its particular use depending on the structure of the function you're analyzing. For instance, differentiating a simple polynomial function might only require the power rule, while the quotient rule is necessary when a function is presented as a fraction of two other functions. Improving your differentiation skills involves a firm grasp of these rules and knowing when to apply them can significantly aid in solving complex calculus problems.

Calculus

Calculus is a branch of mathematics that studies continuous change, which is split into two related sub-fields: differential calculus and integral calculus. Differential calculus focuses on the concept of the derivative, while integral calculus concerns itself with the concept of the integral. Calculus not only helps solve problems involving motion, fluids, and forces but also underpins advanced topics in engineering, statistics, and even computer science.

Mastering calculus requires understanding its fundamental principles and rules, which encompass limits, continuity, derivatives, and integrals. Patience and practice play a significant role in learning calculus efficiently. By breaking down complicated problems into simpler elements, utilizing rules like the quotient rule for differentiation, and integrating these concepts, students can tackle a wide array of problems. Knowledge of calculus is not only academically rewarding but also essential for various scientific and engineering applications.