Question

Use the definition to compute the derivatives of the following functions. $$f(x)=5 x^{2}$$

Short answer

The derivative of \(f(x) = 5x^2\) is \(f'(x) = 10x\).

Step-by-step solution

Understand the power rule for derivatives

The power rule for differentiation states that if we have a function of the form \(f(x) = x^n\), where \(n\) is a constant, then its derivative is \(f'(x) = n\cdot x^{n-1}\). This rule helps us differentiate polynomial functions easily.

Apply the power rule to the function

Given the function \(f(x) = 5x^2\), we identify it in the form \(c\cdot x^n\), where \(c = 5\) and \(n = 2\). According to the power rule, the derivative \(f'(x)\) is found by multiplying the exponent \(n\) by the constant \(c\) and reducing the exponent by 1. Hence, the derivative becomes \(f'(x) = 2 \cdot 5 \cdot x^{2-1}\).

Simplify the expression

Calculating \(2 \cdot 5\), we get \(10\). Thus, \(f'(x)\) simplifies to \(10x^1\), which is simply \(10x\). This is the derivative of the function.

Key concepts

Power Rule

The Power Rule is a fundamental tool in calculus used to find the derivative of polynomials. This rule helps simplify the process of differentiation considerably. The essence of the Power Rule is its application to functions in the form of a constant multiplied by a variable raised to a power, specifically, \(x^n\). When differentiating a function like \(f(x) = x^n\), according to the Power Rule, the derivative is given by \(f'(x) = n \cdot x^{n-1}\). This means you:

• Multiply the exponent \(n\) by the coefficient (if any).
• Subtract one from the exponent to find the new power of \(x\).

For example, consider the function \(f(x) = 5x^2\). Here, the coefficient is \(5\), and the power \(n\) is \(2\). The Power Rule tells us to multiply \(2\) by \(5\) and reduce \(2\) by 1 to get \(f'(x) = 10x^1\), which simplifies to \(10x\). This demonstrates how the Power Rule efficiently gives us the derivative.

Polynomial Functions

Polynomial functions are an important class of functions in mathematics, involving sums of powers of one or more variables multiplied by coefficients. They have the form \(a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0\), where \(a_n, a_{n-1}, ..., a_0\) are constants and \(n\) is a non-negative integer. Key characteristics of polynomial functions include:

• Smooth and continuous curves without breaks or sharp corners.
• Easy to differentiate using the Power Rule.

Differentiation of polynomial functions is usually straightforward due to the Power Rule. Each term in the polynomial can be individually differentiated, and then summed together to find the overall derivative. For example, the derivative of the polynomial \(f(x) = 5x^2\) was calculated as \(f'(x) = 10x\), illustrating how each term's exponent and coefficient are used in the process.

Derivatives

Derivatives are central to the study of calculus, representing the rate at which a function changes at any given point. In more intuitive terms, a derivative is the slope of the tangent line to the function's graph at a specific point. Finding derivatives of functions allows us to understand and predict how one quantity changes with respect to another. The derivative of a function \(f(x)\) is denoted by \(f'(x)\) or \(\frac{df}{dx}\). The process of finding a derivative is known as differentiation. The derivative tells us how the function value changes as \(x\) changes, allowing us to:

• Compute slopes of curves at any point.
• Determine velocity as the rate of change of position.
• Solve optimization problems by finding local maxima and minima.

In the given example, finding the derivative of the function \(f(x) = 5x^2\) utilized the power rule, resulting in \(f'(x) = 10x\). This derivative tells us that the rate of change of \(f(x)\) with respect to \(x\) is proportional to \(x\), with a constant of proportionality of 10.