Problem 57 - Calculus Volume 1

Question

Use the definition of a derivative to find \(f^{\prime}(x)\). $$ f(x)=4 x^{2} $$

Step-by-Step Solution

Answer

The derivative \( f'(x) = 8x \).

Step 1: Recall the Definition of a Derivative

The derivative of a function \(f(x)\), denoted by \(f'(x)\), is defined by the limit: \[ f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} \] This definition will be used to find the derivative of \(f(x) = 4x^2\).

Step 2: Substitute Function into the Derivative Definition

Substitute \(f(x) = 4x^2\) into the derivative formula:\[ f'(x) = \lim_{h \to 0} \frac{4(x+h)^2 - 4x^2}{h} \]

Step 3: Expand the Expression

Expand \((x+h)^2\): \[ (x+h)^2 = x^2 + 2xh + h^2 \] Substitute this back into the limit expression:\[ f'(x) = \lim_{h \to 0} \frac{4(x^2 + 2xh + h^2) - 4x^2}{h} \]

Step 4: Simplify the Expression

Distribute the 4 and simplify:\[ f'(x) = \lim_{h \to 0} \frac{4x^2 + 8xh + 4h^2 - 4x^2}{h} \] Cancel \(4x^2\):\[ f'(x) = \lim_{h \to 0} \frac{8xh + 4h^2}{h} \]

Step 5: Factor and Compute the Limit

Factor \(h\) out from the numerator:\[ f'(x) = \lim_{h \to 0} \frac{h(8x + 4h)}{h} \] Cancel \(h\):\[ f'(x) = \lim_{h \to 0} (8x + 4h) \] Now, as \(h\) approaches 0, the expression becomes:\[ f'(x) = 8x \]

Step 6: Conclusion

Thus, the derivative of the function \(f(x) = 4x^2\) is \( f'(x) = 8x \).

Key Concepts

limit definition of derivative

Understanding the limit definition of a derivative is crucial in calculus. It allows us to determine the rate at which a function's value changes with respect to changes in its input value. The formal definition is given as the limit process: \[ f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} \] Here, \(h\) is an infinitesimally small number that represents a tiny change in \(x\). The expression \(f(x+h) - f(x)\) gives the change in the function's output. The entire fraction represents the average rate of change over a small interval, and taking the limit as \(h\) approaches zero gives the instantaneous rate of change, which is the derivative.

• The numerator \(f(x+h) - f(x)\) captures the change in function value.
• Dividing by \(h\) scales this change per unit change in \(x\).
• The limit \(h \to 0\) refines this to an "instantaneous" change.

This definition can be applied to any differentiable function to find its derivative, offering insights into how the function behaves at any point.

polynomial functions

Polynomial functions, like \(f(x) = 4x^2\), are algebraic expressions consisting of terms in the form \(ax^n\), where \(a\) is a coefficient, and \(n\) is a non-negative integer known as the degree of the term. They have several properties and features that make them one of the fundamental aspects of calculus and algebra.

• Each term of a polynomial is made up of a coefficient and a variable raised to an exponent.
• For \(f(x) = 4x^2\): the term \(4x^2\) has a coefficient 4, and the power of 2 indicates it's a quadratic polynomial.
• Polynomial functions are smooth and continuous, making them easily differentiable.

Polynomials can be used to model a variety of real-world phenomena, from physics to economics. Their simple structure allows for straightforward differentiation and integration, which are pivotal in calculus, helping us understand how changes in variables affect outputs.

differentiation steps

Differentiation involves finding the derivative of a function, and it can be broken down into systematic steps to simplify the process, especially when using the limit definition. Let's recap the differentiation steps used for \(f(x) = 4x^2\):
1. **Substitution:** Quote the function into the derivative formula: \[ f'(x) = \lim_{h \to 0} \frac{4(x+h)^2 - 4x^2}{h} \] 2. **Expansion:** Distribute and expand \((x+h)^2\) to become \(x^2 + 2xh + h^2\). Substitute back: \[ f'(x) = \lim_{h \to 0} \frac{4x^2 + 8xh + 4h^2 - 4x^2}{h} \] 3. **Simplification:** Cancel out like terms \(4x^2\): \[ f'(x) = \lim_{h \to 0} \frac{8xh + 4h^2}{h} \] 4. **Factorization:** Factor out \(h\) from the reduced expression: \[ f'(x) = \lim_{h \to 0} \frac{h(8x + 4h)}{h} \] 5. **Limit Calculation:** Cancel \(h\) and calculate the limit as \(h\) approaches zero: \[ f'(x) = 8x \]
These steps ensure that we find the derivative correctly, making them a powerful technique for evaluating how functions behave at any given point.