Question

Compute the derivative of the given function. $$f(x)=\frac{x^{4}+2 x^{3}}{x+2}$$

Short answer

The derivative is \( f'(x) = 3x^2 \).

Step-by-step solution

Apply the Quotient Rule

Since the function is given as a quotient of two functions, we will apply the quotient rule for derivatives. The quotient rule states that if \( f(x) = \frac{u(x)}{v(x)} \), then the derivative \( f'(x) \) is \( \frac{u'(x) v(x) - u(x) v'(x)}{(v(x))^2} \), where \( u(x) = x^{4} + 2x^{3} \) and \( v(x) = x + 2 \).

Compute the Derivative of the Numerator

Find the derivative of the numerator function \( u(x) = x^{4} + 2x^{3} \). Using the power rule, \( u'(x) = 4x^{3} + 6x^{2} \).

Compute the Derivative of the Denominator

Find the derivative of the denominator function \( v(x) = x + 2 \). Since this is a linear function, \( v'(x) = 1 \).

Substitute into the Quotient Rule Formula

Substitute \( u(x), u'(x), v(x), \) and \( v'(x) \) into the quotient rule formula: \( f'(x) = \frac{(4x^3 + 6x^2)(x + 2) - (x^4 + 2x^3)(1)}{(x + 2)^2} \).

Simplify the Expression

Simplify the expression in the numerator: 1. Expand \((4x^3 + 6x^2)(x + 2)\) to get \(4x^4 + 8x^3 + 6x^3 + 12x^2 = 4x^4 + 14x^3 + 12x^2\).2. Subtract \(x^4 + 2x^3\) from the expanded expression: \(4x^4 + 14x^3 + 12x^2 - x^4 - 2x^3 = 3x^4 + 12x^3 + 12x^2\).Thus, the simplified derivative is: \( f'(x) = \frac{3x^4 + 12x^3 + 12x^2}{(x + 2)^2} \).

Factor the Numerator for Further Simplification (Optional)

Recognize a common factor in the numerator \(3x^2 \) and factor it out to further simplify, if necessary: \( f'(x) = \frac{3x^2(x^2 + 4x + 4)}{(x + 2)^2} \). Notice that \(x^2 + 4x + 4\) is \((x + 2)^2\), allowing for cancellation.

Cancel Common Factors

Cancel the common factor \( (x + 2)^2 \) in the numerator and denominator: \( f'(x) = 3x^2 \).

Key concepts

Derivative of polynomial functions

The derivative of polynomial functions involves understanding how the function's rate of change varies at each point. Polynomials are mathematical expressions consisting of variables and coefficients, combined using addition, subtraction, and multiplication. To find the derivative of a polynomial, we focus on the rate at which this polynomial function changes as its variable changes. This involves applying differentiation techniques like the power rule to determine the derivative with respect to the variable. In the context of our example, we are dealing with a function expressed as a quotient of two polynomial expressions. To compute the derivative, we break down the problem: first differentiating the numerator, then differentiating the denominator, and finally combining these results using the quotient rule. This step-by-step method is vital to ensuring accuracy in differentiating polynomial functions expressed as quotients.

Simplifying derivatives

Simplifying derivatives is an essential step that makes the expression easier to interpret and use. Once the derivative is calculated using rules such as the quotient rule, it's often left in a complicated form. Simplification involves reducing it to its most basic form while maintaining equivalence.In the original solution, the derivative was a fairly complex expression after the initial differentiation. By applying algebraic techniques like expanding expressions and eliminating common factors, we were able to arrive at a much simpler, more manageable result. For instance, factoring common elements such as \(3x^2\) in our example helps simplify the numerator before identifying and canceling common terms in both the numerator and the denominator. This simplification process is crucial for applications like graphing the function or performing further mathematical operations.

Power rule

The power rule is one of the foundational techniques used in differentiation, especially with polynomial functions. It states that if you have a function \(f(x) = x^n\), then the derivative will be \(f'(x) = nx^{n-1}\). This rule is applied to each term in a polynomial function to find its derivative quickly and accurately.In our exercise, applying the power rule to the function's numerator \(u(x) = x^4 + 2x^3\) lets us easily find its derivative, \(u'(x) = 4x^3 + 6x^2\). Each term in the polynomial is handled separately, simplifying the overall process. The power rule is quick and straightforward, emphasizing its utility in dealing with polynomial expressions.

Differentiation techniques

Differentiation techniques encompass a variety of strategies for finding the derivative of different types of functions. Techniques vary depending on the function's structure, whether it’s polynomial, trigonometric, exponential, or a quotient, as in our example. For the given exercise, we apply several differentiation techniques: - **Power Rule**: To differentiate individual terms of the polynomial components. - **Quotient Rule**: To handle the fraction composed of the polynomial in the numerator and a linear function in the denominator. These methods enable us to systematically find the derivative, even for complex functions. Understanding and choosing the appropriate technique ensures efficient computation, accuracy, and easier simplification of derivatives in mathematical analysis.