Problem 36 - College Algebra and Calculus: An Applied Approach

Question

find \(f^{\prime}(x)\). $$ f(x)=x^{2}-3 x-3 x^{-2}+5 x^{-3} $$

Step-by-Step Solution

Answer

The derivative of the function \( f(x)=x^{2}-3 x-3 x^{-2}+5 x^{-3} \) is \( f^\prime(x) = 2x - 3 + 6x^{-3} - 15x^{-4} \).

Step 1: Identify Terms to Differentiate

The function \(f(x) = x^{2} - 3x - 3x^{-2} + 5x^{-3}\) consists of four terms. These are \(x^{2}\), \(-3x\), \(-3x^{-2}\), and \(5x^{-3}\). We will calculate the derivative of each term separately.

Step 2: Apply Power Rule for Derivation

The power rule of derivation states that the derivative of \(x^n\) where \(n\) is a real number, is \(nx^{n-1}\). Let's apply this rule to each term. The derivative of \(x^2\) is \(2x^{2-1} = 2x\), of \(-3x\) is \(-3x^{1-1}= -3\), of \(-3x^{-2}\) is \(-3*(-2)x^{-2-1}=6x^{-3}\), and of \(5x^{-3}\) is \(5*(-3)x^{-3-1} = 15x^{-4}\).

Step 3: Combine the Results

Combine the derivatives we found in the previous step to obtain the derivative of the entire function \(f(x)\). So, \(f^\prime(x) = 2x - 3 + 6x^{-3} - 15x^{-4}\).

Key Concepts

Power Rule of Derivation

The power rule of derivation is a critical tool in calculus, especially when dealing with polynomials. It simplifies the process of finding the rate at which a function changes at any given point. In simple terms, the power rule states that if you have a function of the form f(x) = x^n, where n is a constant real number, the derivative of this function, denoted as f'(x), is nx^{n-1}.

For instance, when applying the power rule to x^2, which is a term of the function in our exercise, we multiply the exponent, 2, by the term and reduce the exponent by one, which gives us 2x^(2-1) or simply 2x. Remember that this rule also applies to negative exponents, as shown by the terms -3x^{-2} and 5x^{-3} in our original function. After applying the power rule, their derivatives become 6x^{-3} and -15x^{-4} respectively.

Calculus

Calculus is the branch of mathematics that studies how things change. It provides a framework for modeling systems in which there is change, and a way to deduce the predictions of such models. With two major branches—differential calculus and integral calculus—calculus is a versatile and powerful tool used in various scientific domains.

Differential calculus focuses on the concept of the derivative, which measures how a function outputs change when the inputs change. Essentially, it's about understanding motion and change at a precise, instantaneous level. Integral calculus, on the other hand, deals with the accumulation of quantities, such as areas under curves and the total growth generated by a continuously growing function. Both branches are interrelated by the fundamental theorem of calculus, forming a coherent whole.

Differentiation

Differentiation is the process of finding the derivative of a function, which is fundamental in calculus. The derivative represents the rate of change of a function concerning its variable. Essentially, differentiation helps us understand how a tiny change in one quantity leads to a change in another.

For practical purposes like our exercise, differentiation enables us to calculate instantaneous rates of change, such as velocity, by finding the slope of a tangent to the curve of a function at any given point. Differentiation can be applied to various types of functions, and it employs different rules, such as the power rule, product rule, quotient rule, and chain rule to tackle more complex functions.

Chain Rule in Calculus

The chain rule is a formula used to compute the derivative of a composite function. In essence, when a function is composed of one function inside another, the chain rule allows us to differentiate the entire composition elegantly and systematically. It says that if you have two functions g(x) and f(x), where g(x) is a function inside f(x), the derivative of the composite function f(g(x)) is the derivative of the outer function evaluated at the inner function multiplied by the derivative of the inner function.

In simpler terms, if h(x) = f(g(x)), then h'(x) = f'(g(x)) * g'(x). This rule is not specifically used in our straightforward exercise, but it's vital for tackling more complicated functions that involve compositions of functions and is often used in conjunction with other rules of differentiation.