Question

In Exercises \(29-32,\) find \(d y / d x\) $$y=4 x^{-2}-8 x+1$$

Short answer

The derivative \(dy/dx\) is \(-8x^{-3} - 8\).

Step-by-step solution

Identify the terms

In this problem, we can identify three terms in the given expression. They are \(4x^{-2}\), \(-8x\), and \(1\). The first term, \(4x^{-2}\), is in the form of \(nx^m\) where \(n = 4\) and \(m = -2\). The second term, \(-8x\), is in the form \(nx^m\) where \(n = -8\) and \(m = 1\). The third term, \(1\), is a constant.

Apply the Power Rule

Applying the power rule to the first term, \(4x^{-2}\), gives the derivative \(-2*4x^{-2-1}\) which simplifies to \(-8x^{-3}\). Applying the power rule to the second term \(-8x\), gives the derivative \(-8x^{1-1}\) which simplifies to \(-8\).

Apply Constant Rule

For the third term \(1\), applying the constant rule, the derivative is \(0\).

Compile and Simplify the Result

The derivative of each term must be compiled into a single expression. So, the derivative of the function \(y = 4x^{-2} - 8x + 1\) is \(-8x^{-3} - 8 + 0\) which simplifies to \(-8x^{-3} - 8\).

Key concepts

Power Rule

When finding the derivative of a function with a term in the form of a power, such as x, we can use the power rule. This rule states that if you have a term of the form , where n is a real number, the derivative is . In simpler terms, we multiply the exponent by the coefficient (the number in front of the term) and then subtract one from the exponent. For example, if we have a term like , its derivative would be , indicating we multiplied the exponent (-2) by the coefficient (4) to get -8 and then subtracted one from the exponent to get -3. The power rule is a fundamental tool for calculus because it simplifies the process of taking derivatives of polynomial functions significantly.

Applying the Power Rule

To apply this rule, identify all terms in the expression that are powers of x. If a term is in the form of , directly apply the power rule to find the derivative of that term. Remember to write the new exponent correctly, as it is now one less than it was before. This is an area where mistakes are often made, so always double-check your exponent after applying the power rule.

Constant Rule

While the power rule applies to terms with variables raised to a power, the constant rule is even simpler. This rule states that the derivative of a constant — a number without any variables — is zero. That's because constants do not change, they have no rate of change, which is effectively what a derivative measures.For example, in our function, there is the constant term 1. Using the constant rule, we know that the derivative of 1 with respect to x is . This rule is vital when simplifying expressions, as it allows us to eliminate constant terms from the derivative expressions, focusing on terms that do actually vary and contribute to the rate of change of the function.

Dealing with Constants

Any constant in your expression will not affect the derivative, other than possibly affecting the y-intercept of the derivative's graph. This makes taking derivatives of functions with several terms less complex, as you can essentially ignore any constants during the differentiation process.

Derivative of a Function

The derivative of a function at a point measures the rate at which the function's value is changing at that point. It's foundational to the study of calculus and explains how quickly values accelerate or decelerate.When finding a derivative of a function like , we typically look at each term individually. We apply rules like the power and constant rules to determine the derivative of each piece of the function. Once each term has been differentiated, we combine these derivatives to form the complete derivative of the function with respect to x.

Understanding the Derivative Conceptually

The concept behind the derivative can be thought of as similar to the speed of a car at a particular moment in time. Just as speed shows how quickly the car's position is changing over time, the derivative shows how quickly the function's output is changing with respect to changes in its input. It's a fundamental concept that allows mathematicians and scientists to make sense of variable change in a precise way.

Simplifying Expressions

After finding the derivatives of individual terms, the next step is simplifying expressions. This process involves combining like terms and consolidating the expression into its most simplified form. Simplifying makes it easier to interpret the derivative and also helps in further applications, such as solving equations or graphing the function's derivative.The simplification step often involves adding or subtracting terms and reducing expressions to remove any unnecessary complexity. For example, if you have a derivative that simplifies to , it is crucial to note that the term can be considered as adding zero and therefore can be omitted for simplicity. The result is a clean, concise derivative that is easier to use in subsequent calculations.

Streamlining the Process

Effective simplification can often require factoring, combining like terms, and recognizing mathematical patterns that can streamline the derivative. Remember that simplification is not just about making an expression shorter; it's about making it clearer and more understandable, which directly correlates to being able to easily grasp and apply the derivative in practical situations.