Question

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

Short answer

The derivative of the function \( f(x) = \frac{4x^3 + 3x^2}{x} \) is \( f'(x) = 8x + 3 \)

Step-by-step solution

Simplify the function

Rewrite the function \( f(x) = \frac{4x^3 + 3x^2}{x} \) in a simpler form by dividing each term in the numerator by \( x \). This gives: \( f(x) = 4x^2 + 3x \)

Apply Power Rule

Now apply power rule to find the derivative of the function. Power rule states that the derivative of \( x^n \) is \( nx^{n-1} \). Applying it gives us \( f'(x) = 8x + 3 \)

Simplify the result

The expression \( f'(x) = 8x + 3 \) is already simplified, so this is the final answer

Key concepts

Power Rule

The power rule is a fundamental tool when finding the derivatives of functions that are formed by polynomial expressions. It states that for any real number exponent, the derivative of the function \( x^n \) where \( n \) is a constant, is \( nx^{n-1} \). This simple yet powerful technique streamlines the process of differentiation by reducing it to a basic operation of multiplying the exponent by the coefficient and then decrementing the exponent by one.

For example, if we have a function such as \( f(x) = x^3 \), applying the power rule would give us \( f'(x) = 3x^{3-1} \), which simplifies to \( f'(x) = 3x^2 \). It is essential to remember that the power rule can be applied to each term in a polynomial expression independently. This not only makes computations straightforward but also significantly speeds up the process of finding derivatives.

Simplifying Expressions

Before applying the power rule, it's often helpful to simplify the expression of the function to make differentiation more manageable. Simplifying expressions involves algebraic manipulations such as factoring, distributing, combining like terms, or in the case of rational functions, dividing the numerator by the denominator if applicable.

For instance, take the function \( f(x) = \frac{4x^3 + 3x^2}{x} \). It's beneficial to first simplify by canceling out common factors. In this case, each term in the numerator shares a common factor of \( x \) with the denominator. Dividing every term by \( x \), we get \( f(x) = 4x^2 + 3x \), a simpler polynomial function to which we can then effortlessly apply the power rule. Always look for opportunities to simplify before differentiating; it’s a place where mistakes can be diminished and calculations can be executed more rapidly.

Derivative of a Function

Understanding the derivative of a function is crucial in calculus. It is a measure of how a function's output value changes as its input value changes. In other words, the derivative represents the rate of change or the slope of the function at a particular point. When we calculate the derivative of a function, we're finding a new function, often denoted as \( f'(x) \), which gives the slope of the original function at any given value of \( x \).

The derivative can provide important information about the behavior of functions: it can tell us where the function is increasing or decreasing, and where it has maxima or minima. Determining the derivative is a key step in many applications, ranging from physics to economics, as it allows us to predict and understand how changes in one quantity lead to changes in another.