Question

In Exercises 39–52, find the derivative of the function. $$ f(x)=\frac{2 x^{4}-x}{x^{3}} $$

Short answer

The derivative of the given function is \(f'(x) = 2 + \frac{2}{x^{3}}\).

Step-by-step solution

Simplify The Function

First, simplify the function by canceling the common factor \(x\) from the numerator and the denominator. Hence the original function \(f(x) = \frac{2x^{4} - x}{x^{3}}\) becomes \(f(x) = 2x - \frac{1}{x^{2}}\).

Differentiate The Simplified Function

Differentiate \(2x\) and \(-\frac{1}{x^{2}}\) individually using the power rule for differentiation. The derivative of \(2x\) is \(2\). Differentiating \(-\frac{1}{x^{2}}\) gives \(2/x^{3}\). Therefore, the derivative of the simplified function \(f'(x)\) is \(2 + \frac{2}{x^{3}}\).

Simplify the Final Result

Lastly, write the final result in the simplest form. The final derivative \(f'(x)\) is \(2 + \frac{2}{x^{3}}\)

Key concepts

Power Rule for Differentiation

Understanding the power rule for differentiation is a cornerstone of calculus. It is a quick and handy method for finding the derivative of a polynomial function. The rule states that if you have a function of the form \( f(x) = x^n \), where \( n \) is a real number, then the derivative of that function \( f'(x) = nx^{n-1} \). This rule simplifies the process of differentiation by reducing it to a simple multiplication and subtraction operation. For example, when differentiating \( x^4 \), the power rule tells us that we should multiply the exponent, 4, by the coefficient of \( x \), which is 1, resulting in 4, and then subtract 1 from the exponent, yielding \( 4x^3 \). This rule is fundamental when tackling calculus exercises and is especially useful when differentiating terms individually before summing up their derivatives to find the derivative of the entire function.

Simplifying Algebraic Expressions

Before finding the derivative of complex functions, it's often beneficial to simplify algebraic expressions. This step can make differentiation much easier and can prevent mistakes. Simplification involves reducing fractions, factoring, and canceling common terms.

For the function \( f(x) = \frac{2x^{4} - x}{x^{3}} \), simplification starts by canceling out a common factor of \( x \) in the numerator and the denominator. After simplification, the function becomes \( 2x - \frac{1}{x^{2}} \). This new expression is simpler to differentiate as it breaks the function down into two separate and more manageable terms. Such a process is crucial because it reveals the underlying structure of the function, which can be directly handled by the standard rules of differentiation.

Finding Derivatives

The derivative of a function at a point is the slope of the tangent line to the function at that point. Finding derivatives is a fundamental task in calculus, often representing the rate at which a quantity is changing. When you simplify the algebraic expression and apply the power rule for differentiation, as we did with function \( f(x) = 2x - \frac{1}{x^{2}} \), we arrive at the derivatives of the individual terms separately: \( 2 \) for \( 2x \) and \( 2/x^{3} \) for \( -1/x^{2} \).

By summing these individual derivatives, we find the overall derivative of the function, which is \( f'(x) = 2 + \frac{2}{x^{3}} \). Remember that the derivative function \( f'(x) \) describes how the value of \( f(x) \) changes as \( x \) changes, which is a key insight into the behavior of the function.

Calculus Exercises

Calculus exercises are designed to reinforce concepts such as finding derivatives and simplifying algebraic expressions. The exercise of finding the derivative of the function \( f(x) = \frac{2x^{4} - x}{x^{3}} \) is a typical calculus problem that tests students' understanding of these concepts. Working through such exercises helps build proficiency in calculus techniques and develops problem-solving skills that are essential for more advanced studies in mathematics, physics, engineering, and related fields.

It's important to practice various types of problems to gain familiarity with different functions and scenarios you might encounter. By consistently applying the power rule for differentiation and simplifying algebraic expressions before finding derivatives, you'll become more adept at tackling the array of calculus exercises you'll face in your studies.