Question

Use the definition of the derivative to compute the derivative of the given function. $$h(x)=x^{3}$$

Short answer

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

Step-by-step solution

Understand the Definition of the Derivative

The derivative of a function \( h(x) \) at a point \( x \) is defined as the limit: \[ h'(x) = \lim_{{h \to 0}} \frac{{h(x+h) - h(x)}}{h}\]This formula gives us the rate of change or the slope of the function \( h(x) \) at any given point \( x \).

Substitute Function into Derivative Definition

For the function \( h(x) = x^3 \), substitute \( x^3 \) into the derivative definition:\[ h'(x) = \lim_{{h \to 0}} \frac{{(x+h)^3 - x^3}}{h}\]

Expand \((x+h)^3\) Using Binomial Theorem

Use the binomial theorem to expand \((x+h)^3\): \[ (x+h)^3 = x^3 + 3x^2h + 3xh^2 + h^3\]

Substitute Expanded Expression into Limit

Replace \((x+h)^3\) in the limit expression:\[ h'(x) = \lim_{{h \to 0}} \frac{{x^3 + 3x^2h + 3xh^2 + h^3 - x^3}}{h}\]The \(x^3\) terms cancel each other out.

Simplify the Expression Inside the Limit

After cancelling \(x^3\), the expression becomes:\[ h'(x) = \lim_{{h \to 0}} \frac{{3x^2h + 3xh^2 + h^3}}{h}\]

Factor Out \(h\) from the Numerator

Factor \(h\) from the numerator:\[ h'(x) = \lim_{{h \to 0}} \frac{{h(3x^2 + 3xh + h^2)}}{h}\]Cancel \(h\) from the numerator and the denominator.

Evaluate the Limit

Cancel \(h\) and evaluate the limit as \(h\) approaches 0:\[ h'(x) = \lim_{{h \to 0}} 3x^2 + 3xh + h^2 = 3x^2\]Thus, the derivative \( h'(x) = 3x^2 \).

Key concepts

Binomial Theorem

The Binomial Theorem is a fundamental principle in algebra that provides a formula to expand expressions that are raised to a power. It is especially useful when dealing with expressions like \((x + h)^n\). This theorem states that: \[(x+y)^n = \sum_{{k=0}}^{n} \binom{n}{k} x^{n-k} y^k\]Where:

• \(n\) is a non-negative integer.
• \(\binom{n}{k}\) represents the binomial coefficients, which can be calculated using factorials: \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\).

This theorem helps break down complex expressions into sums of simpler terms. For example, expanding \((x+h)^3\) with the binomial theorem gives us: \(x^3 + 3x^2h + 3xh^2 + h^3\). This simplification allows for easy substitution into equations, like when finding derivatives.

Limit

Limits are a crucial concept in calculus. They help us understand the behavior of functions as they approach a particular point. The limit examines what value a function gets closer to as its input reaches some value. For example, when we find the derivative of a function, we often use limits to define the rate of change at a specific point.

In our exercise, the derivative \(h'(x)\) is expressed as a limit: \[h'(x) = \lim_{{h \to 0}} \frac{{h(x+h) - h(x)}}{h}\]This shows how \(h(x)\) changes in value as \(h\) approaches zero. The concept of limits is used to transition from average rate of change to instantaneous rate of change, which is what derivatives represent.
This concept is essential in capturing the essence of calculus and analyzing continuous function behavior.

Rate of Change

The rate of change is a measure of how much a quantity changes with respect to another quantity. In mathematics, it often refers to how a function changes as its inputs change.Derivative functions are essentially the rate of change of a function with respect to its variable.
In our example, the derivative of \(h(x) = x^3\), namely \(h'(x) = 3x^2\), tells us the rate at which the value of \(x^3\) changes as \(x\) itself changes.

A positive rate of change indicates that the function is increasing, while a negative rate of change signals a decrease. In real-world contexts, understanding the rate of change allows us to predict trends, such as acceleration in physics, and financial growth in economics.

Slope of a Function

The slope of a function is a concept borrowed from geometry, representing the steepness or incline of a line. In calculus, it takes on a broader meaning. It represents how much a function increases or decreases as its input value changes.

For linear functions, the slope is constant, whereas, for nonlinear functions, the slope can vary at different points. The derivative of a function at a given point gives the slope of the tangent line to the curve at that point.

• A positive slope means the function is rising.
• A negative slope indicates the function is falling.
• A zero slope implies a flat, horizontal tangent, suggesting a local minimum or maximum.

In the given exercise, the function \(h(x)=x^3\) has a derivative \(h'(x)=3x^2\), which provides the slope for any value of \(x\). This equips us with the means to analyze how the function's graph behaves at various points.