Question

Use the definition of the derivative to compute the derivative of the given function. $$g(x)=x^{2}$$

Short answer

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

Step-by-step solution

Understand the Derivative Definition

The derivative of a function \(g(x)\) can be defined by the limit: \( g'(x) = \lim_{h \to 0} \frac{g(x+h) - g(x)}{h} \). Our task is to compute this limit for the function \(g(x) = x^2\).

Substitute the Function into the Derivative Formula

Substitute \(g(x) = x^2\) into the limit definition of the derivative: \( g'(x) = \lim_{h \to 0} \frac{(x + h)^2 - x^2}{h} \).

Expand the Expression

Expand the expression \((x + h)^2\): \((x + h)^2 = x^2 + 2xh + h^2\). Substitute this back into the limit expression: \( g'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - x^2}{h} \).

Simplify the Expression

Cancel out the \(x^2\) terms in the numerator: \( g'(x) = \lim_{h \to 0} \frac{2xh + h^2}{h} \).

Factor and Cancel

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

Evaluate the Limit

Substitute \(h = 0\) into the expression: \( g'(x) = 2x \). Thus, the derivative of \(g(x) = x^2\) is \(2x\).

Key concepts

Limit Definition of Derivative

The limit definition of a derivative is a fundamental concept in calculus. It formally expresses the derivative of a function at a given point. If you think of a function's derivative as its "instantaneous rate of change" or "slope" at a particular moment, you're on the right track! This can be found by calculating a limit. For a function \( g(x) \), the derivative \( g'(x) \) is calculated using the formula:

• \( g'(x) = \lim_{h \to 0} \frac{g(x+h) - g(x)}{h} \)

This formula breaks down to take the difference of the function values at \( x + h \) and \( x \), divide by \( h \), and observe what happens as \( h \) shrinks to zero. This process helps us capture how the function is changing exactly at point \( x \). Understanding this definition is crucial for tackling more advanced topics in calculus. It helps us convert a function's behavior into precise mathematical terms.

Polynomial Function Differentiation

Differentiating polynomial functions using the limit definition is a common task in calculus. Polynomials, such as \( g(x) = x^2 \), have terms that are powers of \( x \). When differentiating such functions, you'll notice a pattern:

• First, substitute the polynomial into the derivative's limit formula. For example, when \( g(x) = x^2 \), you get: \( g'(x) = \lim_{h \to 0} \frac{(x+h)^2 - x^2}{h} \).
• Expand the polynomial expressions where needed. In the \( x^2 \) case, \( (x+h)^2 \) becomes \( x^2 + 2xh + h^2 \).

This systematic approach guides you through step-by-step. Polynomial differentiation within the limit framework showcases the power and simplicity of calculus' foundational tools. As you work with more complex polynomials, this method reveals the consistent structure of their derivatives.

Simplifying Expressions

Simplifying expressions is a vital skill in calculus, especially when dealing with limits. After substituting and expanding the polynomial, you'll often have a more complex expression that needs simplifying. Consider how we've expanded \( (x + h)^2 \) into \( x^2 + 2xh + h^2 \). The goal is to simplify:

• First, cancel any terms that directly negate each other. For the derivative \( g'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - x^2}{h} \), the \( x^2 \) terms simplify.
• What remains is \( \frac{2xh + h^2}{h} \), which can be further simplified by factoring out \( h \), giving \( \frac{h(2x + h)}{h} \).
• Finally, cancel out common factors in the numerator and denominator, leaving us with \( 2x + h \).

These steps might appear minor, but they are essential for reducing expressions to a form where the limit can be comfortably evaluated. Simplification streamlines this process, making derivatives more approachable.

Evaluating Limits

Evaluating limits is the final step in finding a derivative through its limit definition. After simplifying the expression, you'll need to compute the limit as \( h \to 0 \). This involves direct substitution, as in the example expression \( 2x + h \).

• Since we can't directly substitute when there's an \( h \) in the denominator (which is undefined at \( h = 0 \)), simplification is key.
• Once simplified to \( 2x + h \), substitute \( h = 0 \) to find \( g'(x) = 2x \).

Evaluating limits here confirms the derivative of \( g(x) = x^2 \) is \( 2x \). Practicing this direct substitution approach after simplification prepares you for solving more complicated limits with ease. Limits form the backbone of understanding instantaneous change effectively in various contexts.