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 Constant Times a Power Function. $$y=5.4 x^{3}$$

Short answer

The derivative of the function \(y = 5.4x^3\) is \(y' = 16.2x^2\).

Step-by-step solution

Identify the function type

The given function is a product of a constant and a power function. The general rule for differentiation of a function like this is to apply the constant multiple rule and the power rule.

Apply the constant multiple and power rules

To differentiate the function, keep the constant, 5.4, unchanged and multiply it by the derivative of the power function, which involves bringing down the exponent and reducing the exponent by 1.

Calculate the derivative

The derivative of the power function part, \(x^3\), is \(3x^2\). Multiplying this by the constant 5.4 gives the derivative of the entire function.

Write down the derivative

The derivative of \(y = 5.4x^3\) is obtained by combining the constant with the derivative of the power function part: \(y' = 5.4 \cdot 3x^2\).

Simplify the derivative

Simplify the expression by multiplying the constant 5.4 with 3 to find the derivative of the function.

Key concepts

Constant Multiple Rule

In calculus, the constant multiple rule is a fundamental technique used in differentiation. It simplifies the process by allowing us to take constants out of the differentiation operation.
For example, consider a function in the form of \(c \times f(x)\), where \(c\) is a constant, and \(f(x)\) is a differentiable function. The constant multiple rule states that the derivative of this function with respect to \(x\) is \(c\) times the derivative of \(f(x)\) alone:
\[ \frac{d}{dx}[c \times f(x)] = c \times \frac{d}{dx}[f(x)] \]
This rule is especially useful when dealing with functions that include terms multiplied by constants, like \(y = 5.4x^3\) in our exercise. By using the constant multiple rule, we can confidently differentiate the function by focusing on the variable part and later multiplying the result by the constant.

Power Rule

The power rule is another key tool in the differedifferentiation toolbox. It is used when dealing with power functions of the form \(x^n\), where \(n\) is any real number.
The power rule states that for any real number \(n\), the derivative of \(x^n\) is \(n \times x^{n-1}\). Formally, it’s represented as:
\[ \frac{d}{dx}[x^n] = n \times x^{n-1} \]

Applying the Power Rule

When applying the power rule, we simply multiply the exponent by the base raised to one less than the exponent. For example, the derivative of \(x^3\) would be \(3x^2\). This rule greatly simplifies finding derivatives for terms that are powers of \(x\). In our exercise, the power rule was applied to the \(x^3\) term, resulting in \(3x^2\), representing the derivative of the power function part.

Differentiation

At the heart of calculus lies differentiation, a process that calculates the rate at which a function is changing at any given point. It's a cornerstone concept that allows us to determine the slope of the tangent line to the function's curve at any point.
Differentiation is used in a plethora of fields, from physics to economics, whenever there is a need to understand how a quantity changes in response to changes in another quantity.

The Derivative

The derivative of a function, represented by \(f'(x)\) or \(\frac{df}{dx}\), is the limit of the average rate of change as the interval approaches zero. The derivative itself represents the instantaneous rate of change, providing a precise mathematical way to capture the concept of change at any point along the function's domain.

Calculus

Calculus is a branch of mathematics focusing on limits, functions, derivatives, integrals, and infinite series. It's a tool for solving problems in science, engineering, economics, and finance that involve changing quantities.
Calculus has two major branches: differential calculus and integral calculus. Differential calculus is concerned with the rate at which quantities change, while integral calculus deals with the accumulation of quantities.

Real-world Applications

Calculus helps in predicting and understanding phenomena in fields such as physics (motion and forces), biology (population models), economics (maximizing profit or minimizing cost), and even in creating advanced computer algorithms. By understanding both the fundamental principles of calculus and how to apply its techniques, students can solve complex problems that model real-world situations.