Question

In Exercises, find the second derivative of the function. $$ f(x)=x^{2}+7 x-4 $$

Short answer

The second derivative of the function \(f(x) = x^2 + 7x - 4\) is 2.

Step-by-step solution

Compute the first derivative

Use the power rule to find the first derivative. This rule states that \((x^n)' = n*x^{n-1}\). So, the derivative \(f'(x)\) of the function \(f(x) = x^2 + 7x -4\) will be \((2x^1) + (7*(x^0)) = 2x + 7\).

Compute the second derivative

Now, the second derivative \(f''(x)\) is the derivative of the first derivative. We again apply the power rule on the first derivative we found in the previous step. Therefore, the second derivative will be \((2*x^0) + (7*0) = 2+0 = 2\).

Key concepts

Power Rule

In the realm of calculus, the power rule is a fundamental shortcut that makes finding the derivative of a function much simpler. When dealing with polynomial functions like the textbook exercise, the power rule offers a quick way to differentiate terms of the form x^n where n is any real number exponent.

According to the power rule, the derivative of x^n is n*x^(n-1). This rule applies to each term independently, which allows for easy calculation of derivatives term by term. For instance, in the exercise f(x) = x^2 + 7x - 4, applying the power rule results in a first derivative of 2x^(2-1) + 7x^(1-1), simplifying to 2x + 7. Remember, any constant term like -4 drops out as its derivative is zero.

Derivative

The derivative serves as one of the cornerstones of calculus. It measures how a function changes as its input changes---in other words, it's the rate of change or the slope of the function at any given point. Derivatives have immense applications in various fields such as physics, economics, and engineering, where understanding change is crucial.

Calculating a derivative is often referred to as differentiation. For the exercise's function f(x), the first derivative f'(x) represents the slope of f(x) at any point x. To progress further in understanding, it is helpful to practice differentiation on various types of functions to grasp how it represents dynamic change.

Calculus

Calculus is a vast field within mathematics that explores the properties of functions and the behavior of changing quantities. It's divided into two main branches: differential calculus, which deals with the concept of derivatives, and integral calculus, concerned with the concept of integrals.

The role of calculus goes beyond academic exercises; it is essential in modeling and solving real-world problems where variables are constantly changing. For example, in physics, calculus is used to describe motion, while in economics, it's vital for finding optimal solutions. The exercise provided touches on differential calculus, showcasing how calculus is used to understand the nature of mathematical functions in depth.

Function Differentiation

Function differentiation is the process of finding the derivative of a function. It essentially involves applying rules of calculus to discover the instantaneous rate of change of one variable with respect to another.

Differentiation of functions can include applications of various rules, such as the power rule, product rule, quotient rule, and chain rule, depending on the complexity of the function. In the given exercise, differentiating the simple polynomial function f(x) = x^2 + 7x - 4 requires the power rule. The initial differentiation yields the function's first derivative, and differentiating once more, as we do to find the second derivative f''(x), can tell us about the concavity and acceleration of the original function.