Question

Find the derivative of each function. Verify some of your results by calculator. As usual, the letters \(a, b, c, \ldots\) represent constants. Derivative of a Sum. $$y=x^{4}+3 x^{2}+2$$

Short answer

\(y' = 4x^3 + 6x\)

Step-by-step solution

Identify The Given Function

The given function is a sum of three terms: \(y = x^4 + 3x^2 + 2\).

Apply the Sum Rule for Derivatives

According to the sum rule, the derivative of a sum is equal to the sum of the derivatives of each term individually. So, we find the derivatives of \(x^4\), \(3x^2\), and the constant 2 separately.

Differentiate Each Term

Using the power rule, the derivative of \(x^n\) with respect to \(x\) is \(nx^{n-1}\). Thus, the derivative of \(x^4\) is \(4x^3\). The derivative of \(3x^2\) is \(2 \cdot 3x^{2-1} = 6x\). The derivative of a constant is 0.

Combine the Derivatives

Combine the derivatives of the individual terms to find the complete derivative: \(y' = 4x^3 + 6x + 0\).

Simplify the Result

Simplify the expression by eliminating the term with 0, yielding the final derivative: \(y' = 4x^3 + 6x\).

Verification by Calculator

Enter the original function into a calculator that provides symbolic differentiation to check if the computed derivative matches ours. The calculator should confirm that \(y' = 4x^3 + 6x\).

Key concepts

Sum Rule for Derivatives

The sum rule is an essential concept in calculus that states the derivative of a sum of functions is the sum of the derivatives of each individual function. When you have a function that's simply a combination of several terms added together, like the one in our exercise, you can differentiate each term separately and then add up those derivatives to get the final result.

For example, if you have a function like our exercise function, which is a polynomial, you'll apply the sum rule when you break down the function into its components: the fourth power term, the squared term, and the constant term. You'll find the derivative of each of these parts to piece together the solution. It's important that students grasp this rule because it simplifies complex differentiation problems into smaller, more manageable parts.

Power Rule of Differentiation

When you are dealing with functions that involve terms raised to a power, the power rule of differentiation is the go-to tool. This rule states that when you differentiate an expression of the form \( x^n \), where \( n \) is any real number, the derivative is \( nx^{n-1} \).

Applying this to our initial exercise, we differentiate \( x^4 \), which using the power rule becomes \( 4x^3 \). Similarly, we differentiate \( 3x^2 \), where we first consider the constant multiplier separately, leading to \( 6x \). A clear understanding of the power rule enables students to quickly differentiate a wide range of polynomial expressions without much effort.

Symbolic Differentiation

Symbolic differentiation is the process of taking the derivative of a function in terms of symbols rather than specific numbers. In the case of our exercise, we worked through the problem symbolically, first identifying the function components, then applying rules to find the derivative as an expression involving \( x \).

Symbolic differentiation is powerful because it provides a generic formula applicable for different values of \( x \). These generic formulas form the foundation of solutions in physics, engineering, and other disciplines that utilize calculus. Students can also verify their symbolic differentiation results using a calculator with symbolic capabilities as an effective learning check.

Calculus

Calculus is a branch of mathematics that focuses on rates of change (differential calculus) and the accumulation of quantities (integral calculus). It's indispensable for solving problems in physics, engineering, economics, and beyond. In our derivative exercise, we used principles of differential calculus to determine the rate of change of a polynomial function.

Understanding calculus concepts such as derivatives is not only about learning to operate with symbols and numbers. It's about recognizing how these abstract principles describe real-world phenomena, such as how the speed of an object changes over time. For students, the key to mastering calculus involves practice with both the conceptual elements and the various rules and techniques like the sum rule, power rule, and symbolic differentiation.